problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
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8. Let $S_{0}=0$ and $S_{k}=\xi_{1}+\ldots+\xi_{k}, k \geqslant 1$, where $\xi_{1}, \xi_{2}, \ldots$ are independent random variables, $\mathrm{P}\left\{\xi_{k}=1\right\}=p, \mathrm{P}\left\{\xi_{k}=-1\right\}=q, p+q=1$. Show that for $m \leqslant N$ the following equality holds:
$$
\mathrm{P}\left\{\max _{1 \leqslant... | Solution. Since $p^{v} q^{n-v}$ is the probability of a specific trajectory arriving at the point $(n ; m)$, to prove the formula, it is sufficient to show that the number of trajectories of the type
$$
\mathscr{G}_{1}:=\left\{A=\left(S_{0}, \ldots, S_{n}\right): \max _{1 \leqslant k \leqslant n} S_{k} \geqslant N, S_... | proof | Other | proof | Yes | Yes | olympiads | false | 33,577 |
9. Let $\xi_{1}, \xi_{2}, \ldots$ be an infinite sequence of independent Bernoulli random variables, $\mathrm{P}\left\{\xi_{i}=+1\right\}=\mathrm{P}\left\{\xi_{i}=-1\right\}=1 / 2$. Define $S_{0}=0, S_{n}=\xi_{1}+\ldots+\xi_{n}$ and for $x \in Z=\{0, \pm 1, \pm 2, \ldots\}$, define the moments (of the first visit to st... | Solution. (a) The first equality follows from the coincidence of the corresponding events.
(b) Each such trajectory $A=\left(S_{0}, \ldots, S_{n}\right)$, such that $S_{0}0, \ldots, S_{n-1}^{\prime}>0, S_{n}^{\prime}=x$, by symmetry with respect to the point ($n / 2, x / 2$). By relation (5) we have
$$
\mathrm{P}\lef... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 33,578 |
10. Let the conditions of the previous problem be satisfied. Along with the moments $\sigma_{1}(x)$, define the moments
$$
\sigma_{k}(x)=\inf \left\{n>\sigma_{k-1}(x): S_{n}=x\right\}, \quad k=2,3, \ldots
$$
setting $\sigma_{k}(x)=\infty$ if $\{\cdot\}=\varnothing$. (The meaning of these moments is clear: $\sigma_{k}... | Solution. (a) By formula (2) from V1.I. 10, we obtain
$$
\begin{aligned}
& \mathrm{P}\left\{\sigma_{1}(0)=2 n\right\}= \\
& \quad=\mathrm{P}\left\{S_{1} \neq 0, \ldots, S_{2 n-1} \neq 0, S_{2 n}=0\right\}=f_{2 n}=2^{-2 n+1} n^{-1} C_{2 n-2}^{n-1}
\end{aligned}
$$
(b) Using formula (9) from V1.I. 10, we obtain $\mathr... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 33,579 |
11. Let $\xi_{1}, \xi_{2}, \ldots$ be an infinite sequence of independent Bernoulli random variables, $S_{0}=0$ and $S_{n}=\xi_{1}+\ldots+\xi_{n}, n \geqslant 1$. We set
$$
L_{n}(x)=\left|\left\{k \in(0, n]: S_{k}=x\right\}\right| .
$$
(By its meaning, $L_{n}(x)$ is the number of those moments of time $02 n\right\}=2... | Solution. (a) Since the trajectory cannot hit zero on an odd step, $\mathrm{P}\left\{L_{2 n}(0)=k\right\}=\mathrm{P}\left\{L_{2 n+1}(0)=k\right\}$. Note that $2^{-2 n+k} C_{2 n-k}^{n}=\mathrm{P}\left\{S_{2 n-k}=k\right\}$, and we will prove by induction that
$$
\mathrm{P}\left\{L_{2 n}(0)=k\right\}=\mathrm{P}\left\{S_... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 33,580 |
1. Let $\mathscr{D}_{0} \preccurlyeq \mathscr{D}_{1} \preccurlyeq \ldots \preccurlyeq \mathscr{D}_{n}$ be a sequence of partitions, $\mathscr{D}_{0}=\{\Omega\} ; \eta_{k}$ is a $\mathscr{D}_{k}$-measurable random variable, $1 \leqslant k \leqslant n$. Prove that the sequence $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{... | Solution. By the condition, $\eta_{l}-\mathscr{D}_{l}$-measurable quantity, $\mathscr{D}_{l} \preccurlyeq \mathscr{D}_{k}$, $l \leqslant k$. Therefore, $\eta_{l}-\mathscr{D}_{k}$-measurable quantity. Further, $\mathrm{E}\left(\eta_{l} \mid \mathscr{D}_{l-1}\right)-$ $\mathscr{D}_{k}$-measurable quantity for the same re... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,581 |
2. Let the quantities $\eta_{1}, \ldots, \eta_{n}$ be such that $\mathrm{E} \eta_{1}=0$ and $\mathrm{E}\left(\eta_{k} \mid \eta_{1}, \ldots\right.$ $\left.\ldots, \eta_{k-1}\right)=0,1 \leqslant k \leqslant n$. Prove that the sequence $\xi=\left(\xi_{k}\right)_{1 \leqslant k \leqslant n}$,
$$
\xi_{1}=\eta_{1} \quad \t... | Solution. Let $\mathscr{D}_{k}=\mathscr{D}_{\xi_{1}, \ldots, \xi_{k}}$. Since $\left(\xi_{1}, \ldots, \xi_{k}\right)$ is expressed through $\left(\eta_{1}, \ldots, \eta_{k}\right)$, the partition $\mathscr{C}_{k}=\mathscr{D}_{\eta_{1}, \ldots, \eta_{k}}$ is "finer" than the partition $\mathscr{D}_{k}$, i.e., $\mathscr{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,582 |
3. Show that every martingale $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{1 \leqslant k \leqslant n}$ has uncorrelated increments: if $a<b<c<d$, then
$$
\operatorname{cov}\left(\xi_{d}-\xi_{c}, \xi_{b}-\xi_{a}\right)=0
$$ | Solution. Due to the telescoping property of conditional mathematical expectation, the following transformations are valid:
$\xi_{k}=\mathrm{E}\left(\xi_{k+1} \mid \mathscr{O}_{k}\right)=\mathrm{E}\left(\mathrm{E}\left(\xi_{k+2} \mid \mathscr{D}_{k+1}\right) \mid \mathscr{D}_{k}\right)=\mathrm{E}\left(\xi_{k+2} \mid \... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,583 |
4. Let $\xi=\left(\xi_{1}, \ldots, \xi_{n}\right)$ be a random sequence such that $\xi_{k}$ are $\mathscr{D}_{k}$-measurable quantities ( $\mathscr{D}_{1} \preccurlyeq \mathscr{D}_{2} \preccurlyeq \ldots \preccurlyeq \mathscr{D}_{n}$ ). Prove that for this sequence to be a martingale (with respect to the partition syst... | Solution. Necessity. Let $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{1 \leqslant k \leqslant n}$ be a martingale. Then, as shown in the solution to problem I.11.3, $\xi_{k}=\mathrm{E}\left(\xi_{n} \mid \mathscr{D}_{k}\right)$. By definition, $\xi_{\tau}=\sum_{k=1}^{n} \xi_{k} I(\tau=k)$, hence,
$$
\begin{aligned}
& \m... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,584 |
5. Show that if $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{1 \leqslant k \leqslant n}$ is a martingale and $\tau$ is a stopping time, then for any $k \leqslant n$ the following equality holds:
$$
\mathrm{E} \xi_{n} I(\tau=k)=\mathrm{E} \xi_{k} I(\tau=k)
$$ | Solution. Since $\xi_{k}=\mathrm{E}\left(\xi_{n} \mid \mathscr{D}_{k}\right)$ (this was shown in the solution to problem I.11.3), we obtain that
$$
\begin{gathered}
\mathrm{E} \xi_{n} I(\tau=k)=\mathrm{E}\left(\mathrm{E}\left[\xi_{n} I(\tau=k) \mid \mathscr{D}_{k}\right]\right)=\mathrm{E}\left[I(\tau=k) \mathrm{E}\lef... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,585 |
6. Let $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{1 \leqslant k \leqslant n}$ and $\eta=\left(\eta_{k}, \mathscr{D}_{k}\right)_{1 \leqslant k \leqslant n}$ be two martingales, $\xi_{1}=\eta_{1}=0$. Prove that
$$
\mathrm{E} \xi_{n} \eta_{n}=\sum_{k=2}^{n} \mathrm{E}\left(\xi_{k}-\xi_{k-1}\right)\left(\eta_{k}-\eta_{k-... | Solution. Consider the right-hand side of formula (* ):
$$
\begin{aligned}
& \sum_{k=2}^{n} \mathrm{E}\left(\xi_{k}-\xi_{k-1}\right)\left(\eta_{k}-\eta_{k-1}\right)= \\
& \quad=\sum_{k=2}^{n} \mathrm{E}\left(\xi_{k} \eta_{k}\right)-\mathrm{E}\left(\xi_{k} \eta_{k-1}\right)-\mathrm{E}\left(\xi_{k-1} \eta_{k}\right)+\ma... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,586 |
7. Let $\eta_{1}, \ldots, \eta_{n}$ be a sequence of independent and identically distributed random variables, $\mathrm{E} \eta_{i}=0$. Show that the sequences $\xi=\left(\xi_{k}\right)_{1 \leqslant k \leqslant n}$ and $\zeta=\left(\zeta_{k}\right)_{1 \leqslant k \leqslant n}$,
$$
\xi_{k}=\left(\sum_{i=1}^{k} \eta_{i}... | Solution. Since $\xi_{1}, \ldots, \xi_{k}$ are expressed in terms of $\eta_{1}, \ldots, \eta_{k}$, we obtain that
$$
\mathscr{D}_{k} \preccurlyeq D_{\eta_{1}, \ldots, \eta_{k}}=\mathscr{C}_{k} .
$$
We will show that $\mathrm{E}\left(\xi_{k+1} \mid \mathscr{C}_{k}\right)=\xi_{k}$, from which, by the telescoping proper... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,587 |
8. Let $\eta_{1}, \ldots, \eta_{n}$ be a sequence of independent and identically distributed random variables taking values in a (finite) set $Y$. Let $f_{0}(y)=\mathrm{P}\left\{\eta_{1}=y\right\}>0, y \in Y$ and $f_{1}(y)$ be a non-negative function, $\sum_{y \in Y} f_{1}(y)=1$. Show that the sequence
$\xi=\left(\xi, ... | Solution. Since the quantities $\xi_{k}$ are expressed in terms of $\eta_{1}, \ldots, \eta_{k}$, they are $\mathscr{D}_{k}$-measurable. Moreover,
$$
\begin{aligned}
& \mathrm{E}\left(\xi_{k+1} \mid \mathscr{D}_{k}\right)=\mathrm{E}\left(\left.\frac{f_{1}\left(\eta_{1}\right) \ldots f_{1}\left(\eta_{k}\right) f_{1}\lef... | proof | Other | proof | Yes | Yes | olympiads | false | 33,588 |
9. We will say that the sequence
$$
\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{0 \leqslant k \leqslant n}
$$
is a supermartingale (submartingale) if the inequality
$$
\mathrm{E}\left(\xi_{k+1} \mid \mathscr{D}_{k}\right) \leqslant \xi_{k} \quad\left(\geqslant \xi_{k}\right), \quad 0 \leqslant k \leqslant n
$$
holds... | Solution. We will prove the statement for submartingales, then for supermartingales the statement will follow from the fact that if $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{1 \leqslant k \leqslant n}$ is a supermartingale, then $\eta=\left(-\xi_{k}, \mathscr{D}_{k}\right)_{1 \leqslant k \leqslant n}$ is a submarting... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,589 |
10. Let $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{0 \leqslant k \leqslant n}$ and $\eta=\left(\eta_{k}, \mathscr{D}_{k}\right)_{0 \leqslant k \leqslant n}$ be supermartingales and $\tau$ be some stopping time with respect to the partitions $\left(\mathscr{D}_{k}\right)_{0 \leqslant k \leqslant n}$, such that $\mathrm... | Solution. We have $\zeta_{k}=\xi_{k}+\left(\eta_{k}-\xi_{k}\right) I_{\{\tau \leqslant k\}}$. Clearly, $\zeta_{k}$ are $\mathscr{D}_{k}$-measurable quantities. Let's check the inequality $\mathrm{E}\left(\zeta_{k+1}-\zeta_{k} \mid \mathscr{D}_{k}\right) \leqslant 0$.
Indeed,
\[
\begin{aligned}
\mathrm{E}\left(\zeta_{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,590 |
11. Let $\xi=\left(\xi_{k}, \mathscr{D}_{k}\right)_{0 \leqslant k \leqslant n}$ be a submartingale,
$$
\xi_{k}=\sum_{m \leqslant k} I_{A_{m}},
$$
where $A_{m} \in \mathscr{D}_{m}$. Provide the Doob decomposition for this submartingale. | Solution. This is indeed a submartingale, since $\xi_{k}-\mathscr{D}_{k}$-measurable quantities and
$$
\begin{aligned}
\mathrm{E}\left(\xi_{k+1}-\xi_{k} \mid \mathscr{D}_{k}\right)=\mathrm{E}\left(I_{A_{k+1}} \mid \mathscr{D}_{k}\right)=\mathrm{P} & \left(A_{k+1} \mid \mathscr{D}_{k}\right)= \\
= & \sum_{j=1}^{n} \mat... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,591 |
1. Let $\xi=\left(\xi_{0}, \xi_{1}, \ldots, \xi_{n}\right)$ be a Markov chain with values in $X$ and $f=f(x)(x \in X)$ be some function. Will the sequence $\left(f\left(\xi_{0}\right), f\left(\xi_{1}\right), \ldots, f\left(\xi_{n}\right)\right)$ form a Markov chain? Will the "reverse" sequence $\left(\xi_{n}, \xi_{n-1}... | Solution. The sequence $\left(f\left(\xi_{0}\right), f\left(\xi_{1}\right), \ldots, f\left(\xi_{n}\right)\right)$ does not necessarily form a Markov chain. Consider the following example. Let $\xi=\left(\xi_{0}, \xi_{1}, \ldots, \xi_{n}\right)$ be a homogeneous Markov chain with values in $X=\{1 ; 2 ; 3\}$, transition ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,592 |
2. Let $\mathbb{P}=\left\|p_{i j}\right\|, 1 \leqslant i, j \leqslant r,$ be a stochastic matrix and $\lambda$ be an eigenvalue of this matrix, i.e., a root of the characteristic equation det $\|\mathbb{P}-\lambda E\|=0$. Show that $\lambda_{1}=1$ is an eigenvalue, and all other roots $\lambda_{2}, \ldots, \lambda_{r}$... | Solution. The sum of the columns of the matrix $\mathbb{P}-E$ is the zero column,
Suppose there exists an eigenvalue $\lambda$ such that $|\lambda|>1$, and let $\bar{v}$ be the correspondin... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,593 |
3. Let $\xi=\left(\xi_{0}, \xi_{1}, \ldots, \xi_{n}\right)$ be a homogeneous Markov chain with (finite) state space $X$ and transition probability matrix $\mathbb{P}=\left\|p_{x y}\right\|$. Let
$$
T \varphi(x)=\mathrm{E}\left[\varphi\left(\xi_{1}\right) \mid \xi_{0}=x\right] \quad\left(=\sum_{y} \varphi(y) p_{x y}\ri... | Solution. The quantity $\varphi\left(\xi_{k}\right)$ as a function of $\xi_{k}$ is $\mathscr{D}_{k}^{\xi}$-measurable. Further,
$$
\begin{aligned}
& \mathrm{E}\left(\varphi\left(\xi_{k+1}\right) \mid \mathscr{D}_{k}^{\xi}\right)= \\
& \quad=\sum_{i} \mathrm{E}\left(\varphi\left(\xi_{k+1}\right) \mid D_{i}\right) I_{D_... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,594 |
4. Let $\xi=\left(\xi_{n}, \mathbb{H}, \mathbb{P}\right)$ and $\widetilde{\xi}=\left(\widetilde{\xi}_{n}, \widetilde{\mathbb{H}}, \mathbb{P}\right)$ be two Markov chains with different initial distributions $\mathbb{H}=\left(p_{1}, \ldots, p_{r}\right)$ and $\widetilde{\mathbb{H}}=\left(\widetilde{p}_{1}, \ldots, \wide... | Solution. We will prove the statement by induction. For $n=0$, the statement is true:
$$
\sum_{i=1}^{r}\left|\widetilde{p}_{i}-p_{i}\right| \leqslant \sum_{i=1}^{r} \widetilde{p}_{i}+\sum_{i=1}^{r} p_{i}=2
$$
Assume the statement is true for $n=m$. We will prove it for $n=m+1$. Note that $\sum_{i=1}^{r}\left(\widetil... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,595 |
5. Let $P$ and $Q$ be stochastic matrices. Show that $P Q$ and $\alpha P + (1-\alpha) Q, 0 \leqslant \alpha \leqslant 1$, are also stochastic matrices. | Solution. Let $P Q=R=\left(r_{i j}\right)$ and $\alpha P+(1-\alpha) Q=S=\left(s_{i j}\right)$. It is obvious that all $r_{i j}$ and $s_{i j}$ are non-negative. Moreover,
$$
\begin{gathered}
\sum_{j} r_{i j}=\sum_{j} \sum_{k} p_{i k} q_{k j}=\sum_{k} p_{i k} \sum_{j} q_{k j}=\sum_{k} p_{i k}=1 \\
\sum_{j} s_{i j}=\sum_... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,596 |
6. Let $\xi_{0}, \xi_{1}, \ldots, \xi_{N}$ be a Bernoulli sequence of independent random variables, $\mathrm{P}\left\{\xi_{i}=1\right\}=\mathrm{P}\left\{\xi_{i}=-1\right\}=1 / 2$. Define the quantities $\eta_{0}, \eta_{1}, \ldots, \eta_{N}$ by setting $\eta_{0}=\xi_{0}, \eta_{n}=\frac{\xi_{n-1}+\xi_{n}}{2}, 1 \leqslant... | Solution. (a) The sequence $\eta_{0}, \eta_{1}, \ldots, \eta_{n}$, obviously, will be Markov for $N=0,1$. For $N \geqslant 2$ it will not be Markov because, for example,
$$
\begin{aligned}
\mathrm{P}\left\{\eta_{2}=1 \mid \eta_{1}=0, \eta_{0}=-1\right\}=\frac{\mathrm{P}\left\{\eta_{2}=1, \eta_{1}=0, \eta_{0}=-1\right\... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,597 |
7. Let $X_{1}, \ldots, X_{n}$ be i.i.d. random variables. Associate with these variables the rank (order) statistics $X_{1}^{(n)}, \ldots, X_{n}^{(n)}$, obtained by arranging the values $X_{1}, \ldots, X_{n}$ in a non-decreasing order. (So $X_{1}^{(n)}=\min \left(X_{1}, \ldots, X_{n}\right), \ldots$, $X_{n}^{(n)}=\max ... | Solution. We will show that if $X_{i}$ take only two values (for definiteness, 0 and 1), then the order statistics form a Markov chain.
Note that $\mathrm{P}\left\{X_{k-1}^{(n)}=a_{k-1}, \ldots, X_{1}^{(n)}=a_{1}\right\} \neq 0$ if and only if the sequence $\left\{a_{k}\right\}$ is non-decreasing and takes values 0 or... | proof | Other | proof | Yes | Yes | olympiads | false | 33,598 |
1. Let $\Omega=\{r: r \in[0,1] \cap \mathbb{Q}\}, \mathscr{A}$ be an algebra of sets, each of which is a finite sum of non-intersecting sets $A$ of the form $\{r: a<r<b\},\{r: a \leqslant r<b\},\{r: a<r \leqslant b\},\{r: a \leqslant r \leqslant b\}$, and $\mathrm{P}(A)=b-a$. Show that $\mathrm{P}(A), A \in \mathscr{A}... | Solution. The finite additivity of the function $\mathrm{P}$ is obvious. The function P is not countably additive, since
$$
\sum_{r \in[0,1] \cap \mathbb{Q}} \mathrm{P}(\{r\})=0, \quad \text{but} \quad \mathrm{P}(\{r: r \in[0,1] \cap \mathbb{Q}\})=1
$$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,599 |
2. Let $\mathscr{F}$ be the collection of all subsets of some countable set $\Omega$. Define $\mu(A)=0$ if $A$ is finite, and $\mu(A)=\infty$ if $A$ is infinite. Show that the set function $\mu$ is finitely additive but not countably additive. | Solution. The finite additivity of the function $\mu$ is obvious. The function $\mu$ is not countably additive, since if $\Omega=\left\{\omega_{1}, \omega_{2}, \ldots\right\}$, then
$$
\sum_{n} \mu\left(\left\{\omega_{n}\right\}\right)=0 \quad \text { and } \quad \mu(\Omega)=\infty .
$$ | proof | Other | proof | Yes | Yes | olympiads | false | 33,600 |
3. Let $\mu$ be a countably additive measure on $(\Omega, \mathscr{F})$, and let $A_{1}, A_{2}, \ldots \in \mathscr{F}$. Show that
(a) if $A_{n} \uparrow A$, then $\mu\left(A_{n}\right) \uparrow \mu(A)$;
(b) if $A_{n} \downarrow A$ and $\mu\left(A_{m}\right)<\infty$ for some $m$, then $\mu\left(A_{n}\right) \downarro... | Solution. (a) We have $A=\bigcup_{k} B_{k}$, where $B_{1}=A_{1}, B_{k}=A_{k} \backslash A_{k-1}$ for $k>1$. Clearly, $B_{k}$ are pairwise disjoint. Therefore,
$$
\mu(A)=\sum_{k=1}^{\infty} \mu\left(B_{k}\right)=\lim _{n} \sum_{k=1}^{n} \mu\left(B_{k}\right)=\lim _{n} \mu\left(\bigcup_{k=1}^{n} B_{k}\right)=\lim _{n} \... | proof | Other | proof | Yes | Yes | olympiads | false | 33,601 |
5. Let $\mathfrak{J}_{i}, i \in I,$ be arbitrary sets. Establish that the following relation always holds:
$$
\bigcup_{i \in \mathfrak{I}} \bigcap_{j \in \mathfrak{J}_{i}} A_{i j}=\bigcap_{j_{\infty}} \bigcup_{i \in \mathfrak{I}} A_{i j_{i}}
$$
where the intersection is taken over all "paths" $j_{\infty}=\left(j_{i},... | Solution. The relation (*) is easiest to prove by transitioning to the indicators $I_{i j}=I_{i j}(\omega)$ of the sets $A_{i j}$. In this case, its equivalent reformulation is the equality
$$
\max _{i \in \mathcal{J}} \min _{j \in \mathcal{J}_{i}} I_{i j}(\omega)=\min _{j_{\infty}} \max _{i \in \mathcal{J}} I_{i j_{i... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,603 |
6. Show that the "distances" $d(A, B)$ and $\rho(A, B)$, defined by the formulas
$$
\begin{aligned}
& d(A, B)=\mathrm{P}(A \triangle B), \\
& \rho(A, B)= \begin{cases}\frac{\mathrm{P}(A \triangle B)}{\mathrm{P}(A \cup B)}, & \text { if } \mathrm{P}(A \cup B) \neq 0 \\
0, & \text { if } \mathrm{P}(A \cup B)=0\end{cases... | Solution. The triangle inequality for $d$ follows from the inclusion
$$
A \triangle B \subset(A \triangle C) \cup(B \triangle C)
$$
valid for any event $C$.
We will prove the triangle inequality for $\rho$. Fix an arbitrary event $C$. The case when at least one of the events $A \cup B, A \cup C, B \cup C$ has zero p... | proof | Other | proof | Yes | Yes | olympiads | false | 33,604 |
7. Let $\mu$ be a finitely additive measure on the algebra $\mathscr{A}$, the sets $A_{1}, A_{2}, \ldots$ belong to $\mathscr{A}$, are pairwise disjoint, and $A=\sum_{n=1}^{\infty} A_{n} \in \mathscr{A}$. Show that $\mu(A) \geqslant \sum_{n=1}^{\infty} \mu\left(A_{n}\right)$. | Solution. The statement follows from the relations
$$
\mu(A)=\sum_{k=1}^{n} \mu\left(A_{k}\right)+\mu\left(\sum_{k=n}^{\infty} A_{k}\right) \geqslant \sum_{k=1}^{n} \mu\left(A_{k}\right), \quad n \in \mathbb{N}
$$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,605 |
8. Prove that
$$
\begin{aligned}
& \overline{\left(\overline{\lim } A_{n}\right)}=\underline{\lim } \bar{A}_{n}, \quad \overline{\left(\lim A_{n}\right)}=\overline{\lim } \bar{A}_{n}, \\
& \varliminf A_{n} \subseteq \varlimsup A_{n}, \quad \varlimsup\left(A_{n} \cup B_{n}\right)=\varlimsup A_{n} \cup \lim B_{n}, \\
& ... | Solution. All formulas are established by elementary verification taking into account the following definitions: $\overline{\lim } A_{n}$ consists of all outcomes for which an infinite number of events $A_{n}$ have occurred; $\underline{\lim } A_{n}$ consists of all outcomes for which only a finite number of events $A^... | proof | Other | proof | Yes | Yes | olympiads | false | 33,606 |
11. Let $A_{1}, A_{2}, \ldots$ be some sequence of events from $\mathscr{F}$. Show that
$$
\varlimsup A_{n} \backslash \underline{\lim } A_{n}=\varlimsup\left(A_{n} \backslash A_{n+1}\right)=\varlimsup\left(A_{n+1} \backslash A_{n}\right)=\varlimsup\left(A_{n} \triangle A_{n+1}\right)
$$ | Solution. The event $\overline{\lim } A_{n} \backslash \underline{\lim } A_{n}$ is formed by all outcomes $\omega$ that satisfy the following condition: there exist arbitrarily large indices $m, n$ such that $\omega \in A_{n} \backslash A_{m}$ ( $m$ and $n$ are chosen independently). The desired equalities now follow f... | proof | Other | proof | Yes | Yes | olympiads | false | 33,609 |
13. The sequence of events $\left(A_{n}\right)_{n \geqslant 1}$ is called exchangeable (or interchangeable) if the probabilities $p_{n}=\mathrm{P}\left(A_{i_{1}} \ldots A_{i_{n}}\right)$ are the same for any choice of indices $i_{1}<\ldots<i_{n}$ and all $n$. Prove that for such $A_{n}$ the following formulas hold:
\[... | Solution. The first equality is a special case of the inclusion-exclusion formula (see problem I.1.12). Further, we have
$$
\mathrm{P}\left(\lim _{n} A_{n}\right)=\lim _{n} \mathrm{P}\left(\bigcap_{i=n}^{\infty} A_{i}\right)=\lim _{n} \lim _{m} \mathrm{P}\left(\bigcap_{i=n}^{m} A_{i}\right)=\lim _{n} p_{n}=\mathrm{P}\... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 33,611 |
15. Let $\left(A_{n}\right)_{n \geqslant 1}$ be some sequence of sets. Show that
(a) $I\left(\underline{\lim } A_{n}\right)=\underline{\lim } I\left(A_{n}\right), I\left(\overline{\lim } A_{n}\right)=\overline{\lim } I\left(A_{n}\right)$,
(b) $\overline{\lim } I\left(A_{n}\right)-\underline{\lim } I\left(A_{n}\right)... | Solution. (a) The statement follows from the definition of the sets $\lim A_{n}$, $\overline{\lim } A_{n}$.
(b) The statement follows from (a) taking into account the inclusion $\underline{\lim } A_{n} \subseteq$ $\subseteq \widetilde{\lim } A_{n}$.
(c) According to problem II.1.14, we have
$$
I\left(\bigcup_{n} A_{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,613 |
16. Let $\mu$ be a countably additive measure on $(\Omega, \mathscr{F})$. Prove the following Fatou's lemma for sets:
(a) $\mu\left(\underline{\lim } A_{n}\right) \leqslant \underline{\lim } \mu\left(A_{n}\right)$;
(b) if in addition $\mu(\Omega)<\infty$, then
$$
\mu\left(\overline{\lim } A_{n}\right) \geqslant \ove... | Solution. (a) The countable additivity property of the measure $\mu$ is equivalent to the fact that for any increasing sequence of sets $\left(B_{n}\right)_{n \geqslant 1}$, i.e., a sequence such that $B_{n} \subset B_{n+1}$, the equality
$$
\mu\left(\bigcup_{n} B_{n}\right)=\lim _{n} \mu\left(B_{n}\right),
$$
holds,... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,614 |
17. Let $A^{*}=\overline{\lim } A_{n}$ and $A_{*}=\underline{\lim } A_{n}$. Show that
$$
\lim _{n} \mathrm{P}\left(A_{n} \backslash A^{*}\right)=\lim _{n} \mathrm{P}\left(A_{*} \backslash A_{n}\right)=0
$$ | Solution. Since
$$
B_{n}=\bigcap_{k=n}^{\infty} A_{k} \uparrow A_{*} \quad \text { and } \quad \mathrm{P}\left(A_{*} \backslash B_{n}\right) \rightarrow 0, \quad n \rightarrow \infty,
$$
to prove the equality $\lim _{n} \mathrm{P}\left(A_{*} \backslash A_{n}\right)=0$ it suffices to establish that $\lim _{n} \mathrm{... | proof | Other | proof | Yes | Yes | olympiads | false | 33,615 |
20. Show that for any sequences of events $\left(A_{n}\right)$, $\left(B_{n}\right)$, the symmetric difference satisfies the following properties:
$$
\begin{gathered}
A_{1} \triangle B_{1}=\bar{A}_{1} \Delta \bar{B}_{1}, \\
\left(\bigcup_{n} A_{n}\right) \triangle\left(\bigcup_{n} B_{n}\right) \subseteq \bigcup_{n}\le... | Solution. From the equality $\overline{A \triangle B}=A B \cup \bar{A} \bar{B}$, the first relation follows. The second relation is obtained from the easily verifiable inclusion
$$
\left(\bigcup_{n} A_{n}\right) \cap\left(\bigcup_{n} B_{n}\right) \supseteq \bigcup_{n}\left(A_{n} B_{n}\right)
$$
The third relation fol... | proof | Other | proof | Yes | Yes | olympiads | false | 33,618 |
21. Show that for any events $A, B, C$ the inequality
$$
|\mathrm{P}(A B)-\mathrm{P}(A C)| \leqslant \mathrm{P}(B \triangle C) .
$$
holds. | Solution. We have
$$
|\mathrm{P}(A B)-\mathrm{P}(A C)| \leqslant \mathrm{P}(A B \triangle A C) \leqslant \mathrm{P}(B \triangle C)
$$ | proof | Inequalities | proof | Yes | Yes | olympiads | false | 33,619 |
22. Let $\left(A_{n}\right)_{n \geqslant 1}$ be a sequence of events from $\mathscr{F}$ such that
$$
\sum_{n} \mathrm{P}\left(A_{n} \triangle A_{n+1}\right)<\infty
$$
Show that then
$$
\mathrm{P}\left(\overline{\lim } A_{n} \backslash \underline{\lim } A_{n}\right)=0
$$ | Solution. According to the statement of problem II.1.11, we have
$$
\mathrm{P}\left(\overline{\lim } A_{n} \backslash \underline{\lim } A_{n}\right)=\mathrm{P}\left(\overline{\lim } B_{n}\right)
$$
where $B_{n}=A_{n} \triangle A_{n+1}$. Moreover,
$$
\mathrm{P}\left(\overline{\lim } B_{n}\right)=\lim _{n} \mathrm{P}\... | proof | Other | proof | Yes | Yes | olympiads | false | 33,620 |
23. Show that for any events $A$ and $B$ the inequalities
$$
\mathrm{P}(A) \vee \mathrm{P}(B) \leqslant \mathrm{P}(A \cup B) \leqslant \mathrm{P}(A)+\mathrm{P}(B)
$$
and
$$
\mathrm{P}(A \cup B) \mathrm{P}(A \cap B) \leqslant \mathrm{P}(A) \mathrm{P}(B)
$$
hold. When are the equalities achieved? | Solution. Denoting $\mathrm{P}(A \cap B)=p, \mathrm{P}(A \backslash B)=q, \mathrm{P}(B \backslash A)=r$, we can rewrite the inequalities as follows:
$$
p+q \vee r \leqslant p+q+r \leqslant 2 p+q+r, \quad p(p+q+r) \leqslant(p+q)(p+r)
$$
In this form, they are obvious. Equalities in them are achieved respectively at $q... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 33,621 |
24. Let $\left(A_{n}\right)$ and $\left(B_{n}\right)$ be two sequences of events, such that $A_{n} \subseteq B_{n}$ for each $n$. Show that $\overline{\lim } A_{n} \subseteq \overline{\lim } B_{n}$. | Solution. The statement follows from the fact that if $\omega \in \bigcap_{k} A_{n_{k}}$ for some sequence of indices $\left(n_{k}\right)$, then $\omega \in \bigcap_{k} B_{n_{k}}$. | proof | Other | proof | Yes | Yes | olympiads | false | 33,622 |
25. Let $\left(A_{n}\right)_{n \geqslant 1}$ and $\left(B_{n}\right)_{n \geqslant 1}$ be two sequences of events, such that
$$
\mathrm{P}\left(\overline{\lim } A_{n}\right)=1, \quad \mathrm{P}\left(\overline{\lim } \bar{B}_{n}\right)=0
$$
Show that $\mathrm{P}\left(\overline{\lim } A_{n} B_{n}\right)=1$. | Solution. According to problem II.1.8, we have
$$
\varlimsup A_{n}=\varlimsup A_{n} B_{n} \cup \varlimsup A_{n} \bar{B}_{n} .
$$
By problem II.1.24, we obtain $\mathrm{P}\left(\overline{\lim } A_{n} \bar{B}_{n}\right)=0$, from which we find that $\mathrm{P}\left(\overline{\lim } A_{n} B_{n}\right)=1$. | proof | Other | proof | Yes | Yes | olympiads | false | 33,623 |
26. Provide an example of two probability measures $\mathrm{P}$ and $\mathrm{Q}$, for which the smallest measure $\nu$ with the properties $\nu \geqslant \mathrm{P}$ and $\nu \geqslant \mathrm{Q}$ is not $\mathrm{P} \vee \mathrm{Q}$ $(=\max \{P, Q\})$, but $P+Q$. | Solution. Let $\Omega=\{0,1\}, \mathrm{P}(\{0\})=1, \mathrm{Q}(\{1\})=1$. Then $\nu=\mathrm{P}+\mathrm{Q}$, since $\nu(\{0\}), \nu(\{1\}) \geqslant 1$ and measure $\nu$ is the smallest, however $\nu(\Omega)=2 \neq 1=$ $=\mathrm{P}(\Omega) \vee \mathrm{Q}(\Omega)$. | proof | Other | math-word-problem | Yes | Yes | olympiads | false | 33,624 |
27. Let on a measurable space $(\Omega, \mathscr{F})$ be given a sequence of probability measures $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots$, such that for each $A \in \mathscr{F}$
$$
\mathrm{P}_{n}(A) \rightarrow \mathrm{P}(A),
$$
where $\mathrm{P}(A)$ is some set function $A \in \mathscr{F}$. Prove the Vitali-Hahn-Sa... | Solution. (a) The validity of all properties of a probability measure for $\mathrm{P}$, except for countable additivity, follows from the properties of the limit. Countable additivity is equivalent to continuity at zero (since $\mathrm{P}(\Omega)<\infty$) and follows from part (b).
(b) Let $\lim _{k} \sup _{n} \mathrm... | proof | Other | proof | Yes | Yes | olympiads | false | 33,625 |
28. Construct on $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$ measures $\nu_{1}, \nu_{2}, \ldots$ such that the sequence $\left(\nu_{n}(B)\right)_{n \geqslant 1}$ is non-increasing for each $B \in \mathscr{B}(R)$ and such that the limit function of sets $\nu(B)=\lim _{n} \nu_{n}(B), B \in \mathscr{B}(R)$, is not a (countabl... | Solution. Consider $\nu_{n}(B)=\lambda(B \backslash[-n, n])$, where $\lambda$ is the Lebesgue measure. The sequence $\nu_{n}(B)$ is non-increasing in $n$, and
$$
\nu_{n}(B)=\infty \Leftrightarrow \lambda(B)=\infty
$$
In this case, $\nu(B)=\infty$ if $\lambda(B)=\infty$ and 0 otherwise. Clearly, $\nu$ is not a countab... | proof | Other | proof | Yes | Yes | olympiads | false | 33,626 |
29. Let $\Omega$ be a set consisting of no more than a countable number of elements, and let $\mathscr{F}$ be some $\sigma$-algebra of its subsets. Show that there always exists a partition $D_{1}, D_{2}, \ldots$ (i.e., $\bigcup_{n} D_{n}=\Omega$ and $D_{m} \cap D_{n}=\varnothing, m \neq n$), generating $\mathscr{F}$:
... | Solution. The corresponding partition can be constructed by considering on $\Omega$ the equivalence classes formed by the relation “ $\omega_{1} \sim \omega_{2} »$, defined as follows:
$$
\omega_{1} \sim \omega_{2} \Leftrightarrow\left(\forall A \in \mathscr{F}: \omega_{1} \in A \Leftrightarrow \omega_{2} \in A\right)... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 33,627 |
1. Let $\mathscr{B}_{1}$ and $\mathscr{B}_{2}$ be $\sigma$-algebras of subsets of a space $\Omega$. Will the systems of sets
$$
\begin{aligned}
& \mathscr{B}_{1} \cap \mathscr{B}_{2} \equiv\left\{A: A \in \mathscr{B}_{1} \text { and } A \in \mathscr{B}_{2}\right\}, \\
& \mathscr{B}_{1} \cup \mathscr{B}_{2} \equiv\left... | Solution. By direct verification, we obtain that $\mathscr{B}_{1} \cap \mathscr{B}_{2}$ is a $\sigma$-algebra. The system of sets $\mathscr{B}_{1} \cup \mathscr{B}_{2}$ is a $\sigma$-algebra if and only if $\mathscr{B}_{1} \subseteq \mathscr{B}_{2}$ or $\mathscr{B}_{2} \subseteq \mathscr{B}_{1}$. Indeed, let $\mathscr{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,629 |
2. Let $\mathscr{D}=\left\{D_{1}, D_{2}, \ldots\right\}$ be some countable partition of $\Omega\left(D_{n} \neq \varnothing\right)$. What is the cardinality of the $\sigma$-algebra $\sigma(\mathscr{D})$? | Solution. The sigma-algebra $\sigma(\mathscr{D})$ contains a continuum of different elements, which can be verified by associating each sequence $\alpha=\left(\alpha_{1}, \alpha_{2}, \ldots\right)$, consisting of zeros and ones, with the set $D^{\alpha}=D_{1}^{\alpha_{1}} \cup D_{2}^{\alpha_{2}} \cup \ldots$, where $D_... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,630 |
3. Show that
$$
\mathscr{B}\left(\mathbb{R}^{n}\right) \otimes \mathscr{B}(\mathbb{R})=\mathscr{B}\left(\mathbb{R}^{n+1}\right)
$$ | Solution. The set $G_{n} \times G_{1} \subseteq \mathbb{R}$ is open in $\mathbb{R}^{n+1}$ when $G_{n}$ and $G_{1}$ are open sets in $\mathbb{R}^{n}$ and $\mathbb{R}$ respectively, hence $\mathscr{B}\left(\mathbb{R}^{n}\right) \otimes \mathscr{B}(\mathbb{R}) \subseteq \mathscr{B}\left(\mathbb{R}^{n+1}\right)$. Conversel... | proof | Other | proof | Yes | Yes | olympiads | false | 33,631 |
4. Prove that the following sets belong to $\mathscr{B}\left(\mathbb{R}^{\infty}\right)$:
$$
\left\{x: \varlimsup_{n} x_{n} \leqslant a\right\}, \quad\left\{x: \lim _{n} x_{n}>a\right\}
$$
$\left\{x: x_{n} \rightarrow\right\}$ - the set of those $x$ for which $\lim _{n} x_{n}$ exists and is finite,
$$
\left\{x: \sum... | Solution. The belonging of the first set to $\mathscr{B}\left(\mathbb{R}^{\infty}\right)$ follows from the formula
$$
\left\{x: \overline{\lim _{n}} x_{n} \leqslant a\right\}=\bigcap_{k=1}^{\infty} \bigcup_{m=1}^{\infty} \bigcap_{n=m}^{\infty}\left\{x: x_{n}<a+\frac{1}{k}\right\}
$$
Verbal expression of this formula:... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,632 |
5. Prove that the following sets do not belong to $\mathscr{B}\left(\mathbb{R}^{[0,1]}\right)$:
$$
\left\{x: x_{t}=0 \text { for at least one } t \in[0,1]\right\} \text {, }
$$
$\left\{x\right.$ : the function $x_{t}$ is continuous at a fixed point $\left.t_{0} \in[0,1]\right\}$, where $x=\left(x_{t}\right)_{0 \leqsl... | Solution. All sets in $\mathscr{B}\left(\mathbb{R}^{[0,1]}\right)$ are defined by the "behavior" of functions at a countable number of points. In particular, for any $B \in \mathscr{B}\left(\mathbb{R}^{[0,1]}\right)$, there always exist $t_{1}, t_{2}, \ldots \in[0,1]$ such that if $x=\left(x_{t}\right) \in B$ and $y_{t... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,633 |
6. Let $D-$ be the space of all functions $x=\left(x_{t}\right), t \in[0,1]$, which are right-continuous for all $t0$. Prove that the function
$$
d(x, y)=\inf _{\lambda \in \Lambda}\left[\sup _{t}\left|x_{t}-y_{\lambda(t)}\right|+\sup _{t}|t-\lambda(t)|\right]
$$
defines a metric (Skorokhod) on $D$, where $\Lambda$ i... | Solution. Since $\lambda^{-1} \in \Lambda$, and
$$
\sup _{t}\left|u_{t}-v_{\lambda(t)}\right|=\sup _{t}\left|u_{\lambda^{-1}(t)}-v_{t}\right|
$$
for $\left(u_{t}, v_{t}\right)=\left(x_{t}, y_{t}\right)$ or $\left(u_{t}, v_{t}\right)=(t, t)$, we obtain that
$$
d(x, y)=d(y, x)
$$
Let's check the triangle inequality. ... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,634 |
7. Prove that $\mathscr{B}_{0}\left(\mathbb{R}^{d}\right)=\mathscr{B}\left(\mathbb{R}^{d}\right), d \geqslant 1$, and $\mathscr{B}_{0}\left(\mathbb{R}^{\infty}\right)=\mathscr{B}\left(\mathbb{R}^{\infty}\right)$. | Solution. In the space $\mathbb{R}^{d}$, any ball can be represented as a countable union of cubes whose edges are parallel to the corresponding coordinate axes. Therefore, all balls generating $\mathscr{B}_{0}\left(\mathbb{R}^{d}\right)$ lie in $\mathscr{B}\left(\mathbb{R}^{d}\right)$, so $\mathscr{B}_{0}\left(\mathbb... | proof | Other | proof | Yes | Yes | olympiads | false | 33,635 |
8. By definition, a system $\mathscr{I}$ of subsets of $\Omega$ is called a $\lambda$-system if
(a) $\Omega \in \mathscr{I}$;
(b) $(A, B \in \mathscr{I}$ and $A \subseteq B) \Rightarrow B \backslash A \in \mathscr{I}$;
(c) $\left(A_{n} \in \mathscr{I}, n \geqslant 1\right.$, and $\left.A_{n} \uparrow A\right) \Right... | Solution. Let conditions (a), (b), and (c) be satisfied. We will show that conditions ( $b^{\prime}$ ) and ( $c^{\prime}$ ) also hold. From the relations $\bar{B}=\Omega \backslash B$ and $\Omega \in \mathscr{I}$, condition (b') follows. Also, note that if $A, B \in \mathscr{I}$ and $A \cap B=\varnothing$, then $A \cup... | proof | Other | proof | Yes | Yes | olympiads | false | 33,636 |
9. The theorem on monotone classes states that $\mu(\mathscr{A})=\sigma(\mathscr{A})$ for any algebra of sets $\mathscr{A}$. Using this fact, establish the following statement: if $\mathscr{G}$ is a $\pi$ - $\lambda$-system, then $\mathscr{G}$ is a $\sigma$-algebra. | Solution. The system $\mathscr{G}$ is closed under intersections (since it is a $\pi$-system) and complements (since it is a $\lambda$-system), so $\mathscr{G}$ is an algebra. Moreover, $\mathscr{G}$ is a monotone class, as all $\lambda$-systems are. Indeed, if $A_{n} \in \mathscr{G}, n \geqslant 1$, and $A_{n} \uparro... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,637 |
10. Let $\mathscr{I}=\left\{A \in \mathscr{F}: I_{A} \in \mathscr{H}\right\}$, where $\mathscr{H}$ is some system of $\mathscr{F}$-measurable functions satisfying the following properties:
(a) if $f, g \in \mathscr{H}$, then $f+g \in \mathscr{H}$ and $c f \in \mathscr{H}$ for any $c \in \mathbb{R}$;
(b) if $h_{n} \in... | Solution. From properties (a) and (b) it follows that
1) if $I_{A}, I_{B} \in \mathscr{H}$ and $A \subseteq B$, then $I_{B \backslash A}=I_{B}+(-1) I_{A} \in \mathscr{H}$;
2) if $I_{A_{n}} \in \mathscr{H}, n \geqslant 1$ and $I_{A_{n}} \uparrow h$, then $h=I_{\bigcup_{n=1}^{\infty} A_{n}} \in \mathscr{H}$.
Having sta... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,638 |
11. It is said that a $\sigma$-algebra is countably generated or separable if it is generated by some countable class of subsets.
(a) Show that the $\sigma$-algebra $\mathscr{B}(\mathbb{R})$ of Borel sets in $\mathbb{R}$ is countably generated.
(b) Provide an example showing that it is possible for two $\sigma$-algeb... | Solution. (a) The $\sigma$-algebra $\mathscr{B}(\mathbb{R})$ is generated by sets of the form $(a, b], a, b \in \mathbb{Q}$.
(b) Take $\mathscr{F}_{2}=\mathscr{B}(\mathbb{R})$, and let $\mathscr{F}_{1}$ be all at most countable sets in $\mathscr{B}(\mathbb{R})$. | proof | Other | proof | Yes | Yes | olympiads | false | 33,639 |
12. Let $(\Omega, \mathscr{F})$ be a measurable space, where the $\sigma$-algebra $\mathscr{F}$ is countably generated by some countable partition $\mathscr{D}=\left\{D_{1}, D_{2}, \ldots\right\}$. Show that the $\sigma$-algebra $\mathscr{F}$ coincides with the $\sigma$-algebra
$$
\sigma(X)=\left\{X^{-1}(B): B \in \ma... | Solution. The random variable is obviously measurable with respect to $\mathscr{F}$ (as a Borel function of the indicators $I_{D_{n}}$). Therefore, $\sigma(X) \subseteq \mathscr{F}$. Conversely,
$D_{n}=\{n$-th digit in the decimal expansion of $X$ is 5$\}$, hence $\mathscr{F}=\sigma\left(D_{1}, D_{2}, \ldots\right) \s... | proof | Other | proof | Yes | Yes | olympiads | false | 33,640 |
13. Show that a $\sigma$-algebra $\mathscr{G}$ is countably generated if and only if $\mathscr{G}=\sigma(X)$ for some random variable $X$. | Solution. If $\mathscr{G}=\sigma(X)$, then $\mathscr{G}=\sigma(\{a<X \leqslant b\}: a, b \in \mathbb{Q})$. Conversely, if $\mathscr{G}=\sigma\left(A_{n}, n \geqslant 1\right)$, then let
$$
X=\sum_{n=1}^{\infty} 10^{-n}\left(2 I_{A_{n}}+3\right)
$$
Then proceed as in the solution of problem II.2.12. | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,641 |
14. Let $(\Omega, \mathscr{F}, \mathrm{P})$ be a complete probability space (see problem II.2.36 for more details), $\mathscr{G}$ a $\sigma$-subalgebra of $\sigma$-algebra $\mathscr{F}$, and $\mathscr{E}_{1} \supseteq \mathscr{E}_{2} \supseteq \ldots$ a sequence of $\sigma$-subalgebras of $\sigma$-algebra $\mathscr{F}$... | Solution. (a) We will show that the quantity $\xi_{0}$ is measurable with respect to $\bigcap_{n} \sigma\left(\mathscr{G}, \mathscr{E}_{n}\right)$, and thus $\bigcap_{n} \sigma\left(\mathscr{G}, \mathscr{E}_{n}\right)=\sigma\left(\mathscr{G}, \xi_{0}\right)$. At the same time, $\bigcap_{n} \mathscr{E}_{n}=\{\varnothing... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,642 |
15. Let $\mathscr{L}$ be a $\lambda$-system of subsets of $\Omega$. Then if $A, B \in \mathscr{L}$ and $A \cap B = \varnothing$, then $A \cup B \in \mathscr{L}$. | Solution. By the definition of a $\lambda$-system, $\Omega \backslash A=B \cup C \in \mathscr{L}$ when $C=\Omega \backslash(A \cup B)$. Moreover, $C=(B \cup C) \backslash B \in \mathscr{L}$. Finally, we obtain $A \cup B=\Omega \backslash C \in \mathscr{L}$. | proof | Other | proof | Yes | Yes | olympiads | false | 33,643 |
16. Let $\mathscr{A}$ and $\mathscr{B}$ be two $\sigma$-algebras of subsets of $\Omega$. Define
$$
d(\mathscr{A}, \mathscr{B})=4 \text { sup }\{|\mathrm{P}(A \cap B)-\mathrm{P}(A) \mathrm{P}(B)|: A \in \mathscr{A}, B \in \mathscr{B}\} .
$$
Show that this quantity, characterizing the degree of dependence between $\mat... | Solution. (a) Since
$$
\mathrm{P}(A)=\mathrm{P}(A B)+\mathrm{P}(A \bar{B}), \quad \mathrm{P}(B)=\mathrm{P}(\bar{A} B)+\mathrm{P}(A B)
$$
we obtain
$$
|\mathrm{P}(A B)-\mathrm{P}(A) \mathrm{P}(B)| \leqslant \sup |x-(x+y)(x+z)|=\frac{1}{4},
$$
where the upper bound is taken over all $x, y, z \in[0,1]$, the sum of whi... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,644 |
17. Prove the existence and uniqueness of the classes $\lambda(\mathscr{E})$ and $\pi(\mathscr{E})$, containing the system $\mathscr{E}$ of subsets of the set $\Omega$. | Solution. The set of all $\pi$-systems containing $\mathscr{E}$ is non-empty, as it contains the set of all subsets of $\Omega$. Then the intersection of all $\pi$-systems is non-empty (since $\mathscr{E}$ lies in all of them) and is the smallest $\pi$-system containing $\mathscr{E}$. The same is true for $\lambda$-sys... | proof | Other | proof | Yes | Yes | olympiads | false | 33,645 |
18. Let $\mathscr{A}$ be some algebra of sets, possessing the property that any sequence $\left(A_{n}\right)_{n \geqslant 1}$ of pairwise disjoint sets $A_{n} \in \mathscr{A}$ is such that $\bigcup_{n=1}^{\infty} A_{n} \in \mathscr{A}$. Prove that then $\mathscr{A}$
is a $\sigma$-algebra. | Solution. The statement is an elementary consequence of the equality
$$
\bigcup_{n=1}^{\infty} B_{n}=\bigcup_{n=1}^{\infty} A_{n}
$$
where $A_{n}$ are pairwise disjoint sets defined by the relations
$$
A_{1}=B_{1}, \quad A_{2}=B_{2} \backslash B_{1}, \quad A_{3}=B_{3} \backslash\left(B_{1} \cup B_{2}\right),
$$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,646 |
19. (See [91].) Let $\left(\mathscr{A}_{n}\right)_{n \geqslant 1}$ be a strictly increasing sequence of algebras, i.e., $\mathscr{A}_{n} \subset \mathscr{A}_{n+1}, n \geqslant 1$. Show that $\mathscr{A}=\bigcup_{n} \mathscr{A}_{n}$ is an algebra.
It is not difficult to provide an example of $\sigma$-algebras $\mathscr... | Solution. Let's check that $\mathscr{A}=\bigcup_{n} \mathscr{A}_{n}$ is an algebra. If $A \in \mathscr{A}$, then $A \in \mathscr{A}_{n}$ starting from some $n$ and, consequently, $\bar{A} \in \mathscr{A}_{n} \subset \mathscr{A}$. Similarly, it is established that $A \cup B \in \mathscr{A}$ when $A, B \in \mathscr{A}$. ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,647 |
20. Let $\mathscr{F}$ be a $\sigma$-algebra, and let the set $C$ not belong to $\mathscr{F}$. Consider the $\sigma$-algebra $\sigma(\mathscr{F} \cup\{C\})$ generated by $\mathscr{F}$ and $C$. Show that
$$
\sigma(\mathscr{F} \cup\{C\})=\{A C \cup B \bar{C}: A, B \in \mathscr{F}\}
$$
Prove a similar property in the cas... | Solution. The statement follows from the formulas
$$
\begin{gathered}
\overline{A C \cup B \bar{C}}=\bar{A} C \cup \bar{B} \bar{C} \\
\bigcup_{n}\left(A_{n} C \cup B_{n} \bar{C}\right)=\left(\bigcup_{n} A_{n}\right) C \cup\left(\bigcup_{n} B_{n}\right) \bar{C}
\end{gathered}
$$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,648 |
22. Let $C[0,1]$ be the space of continuous functions
$$
x=\{x(t)\}_{0 \leqslant t \leqslant 1} \quad \text { with the metric } \rho(x, y)=\sup _{0 \leqslant t \leqslant 1}|x(t)-y(t)| \text {. }
$$
Prove that the space $C[0,1]$ is Polish, i.e., a complete separable metric space. | Solution. Due to the completeness of $\mathbb{R}$, any fundamental sequence $x_{n} \in C[0,1]$ converges uniformly to some function $x$. We will show that $x \in C[0,1]$. We have
$$
|x(t)-x(s)| \leqslant\left|x_{n}(t)-x_{n}(s)\right|+2 \rho\left(x_{n}, x\right)
$$
By choosing a sufficiently large $n$, we ensure that ... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,650 |
23. Let $C=C[0, \infty)$ be the space of continuous functions $x=$ $=\left(x_{t}\right)$ defined for $t \geqslant 0$. Show that with respect to the metric
$$
\rho(x, y)=\sum_{n=1}^{\infty} 2^{-n} \min \left\{\sup _{0 \leqslant t \leqslant n}\left|x_{t}-y_{t}\right|, 1\right\}, \quad x, y \in C
$$
this space is Polish... | Solution. We will show that the metric space $C$ is complete. Let the sequence $x^{n}=\left(x_{t}^{n}\right), n \geqslant 1$, be fundamental in $C$. Then for each $m=1,2, \ldots$ the functions $\left.x^{n}\right|_{[0, m]}=\left(x_{t}^{n}\right)_{0 \leqslant t \leqslant m}$ form a fundamental sequence in the Polish spac... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,651 |
24. Let $\mathscr{A}$ be a non-empty system of subsets of a set $\Omega$. Show that the algebra $\alpha(\mathscr{A})$ generated by this system can be constructed as follows.
Set $\mathscr{A}_{1}=\mathscr{A} \cup\{\varnothing, \Omega\}$ and
$$
\mathscr{A}_{n+1}=\left\{A \cup \bar{B}: A, B \in \mathscr{A}_{n}\right\} \... | Solution. Obviously, $\mathscr{A}_{n} \subseteq \alpha(\mathscr{A})$, therefore $\bigcup_{n} \mathscr{A}_{n} \subseteq \alpha(\mathscr{A})$. Moreover, if $A \in \mathscr{A}_{m}$, then $\bar{A} \in \mathscr{A}_{m+1}$, and if $B \in \mathscr{A}_{n}$, then $A \cup B \in \mathscr{A}_{n \vee m+2}$. In other words, $\bigcup_... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,652 |
26. Let us call a body a system of sets that is closed under the operations of countable union and intersection. Verify that the $\sigma$-algebra of Borel sets $\mathscr{B}(\mathbb{R})$ is a body $\mathscr{B}$ generated by open sets. | Solution. Any closed set in $\mathbb{R}$ is a countable intersection of some open sets, so $\mathscr{B}$ contains all closed sets. To complete the proof, one should use the principle of suitable sets, establishing that the body $\mathscr{B}$ is closed under complements, and thus is a $\sigma$-algebra, coinciding with $... | proof | Other | proof | Yes | Yes | olympiads | false | 33,654 |
27. The structure of the $\sigma$-algebra generated by an infinite system of sets $\mathscr{A}$ can be quite complex (see Problem II.2.25). However, in some cases, one can explicitly specify a system $S(\mathscr{A})$, somewhat larger than $\sigma(\mathscr{A})$, and having the same cardinality as $\sigma(\mathscr{A})$.
... | Solution. Let $A^{m_{1} \ldots m_{l}} \in S(\mathscr{A})$ for all $l, m_{1}, m_{2}, \ldots \geqslant 1$ and, therefore, for some $A_{n_{1} \ldots n_{k}}^{m_{1} \ldots m_{l}} \in \mathscr{A}$, the equality holds
$$
A^{m_{1} \ldots m_{l}}=\bigcup_{n_{\infty}} \bigcap_{k=1}^{\infty} A_{n_{1} \ldots n_{k}}^{m_{1} \ldots m... | proof | Other | proof | Yes | Yes | olympiads | false | 33,655 |
28. Let $\mathfrak{c}$ be the power of the continuum. Show that the power of Borel sets in $\mathbb{R}$ is $\mathfrak{c}$, and the power of the $\sigma$-algebra of Lebesgue measurable sets is $2^{\mathfrak{c}}$. | Solution. The number of Borel sets is the continuum. This follows from problems II.2.11, I. 2.26, and the remark to problem II.2.27.
Since the Cantor set $\mathscr{C}$ (see problem II.3.20) has measure zero, any of its subsets is Lebesgue measurable. The number of such subsets is $2^{\text{c}}$, since $|\mathscr{C}|=c... | proof | Other | proof | Yes | Yes | olympiads | false | 33,656 |
29. Show that there does not exist a $\sigma$-algebra with the power of the set of natural numbers $\aleph_{0}$, consisting of a countable number of elements. Thus, the structure of $\sigma$-algebras is such that they always either consist of a finite number of elements or have an uncountable power. According to proble... | Solution. Let $\left\{A_{n}\right\}_{n \geqslant 1}$ be an infinite number of distinct elements of the $\sigma$-algebra $\mathscr{A}$. For any sequence $\alpha=$ $=\left(\alpha_{1}, \alpha_{2}, \ldots\right)$, consisting of zeros and ones, form all possible intersections
$$
D_{\alpha}=\bigcap_{n=1}^{\infty} A_{n}^{\al... | proof | Other | proof | Yes | Yes | olympiads | false | 33,657 |
31. The construction and conclusion of Problem II. 2.25 show that the structure of $\sigma$-algebras can be quite complex. In 1916, M. Ya. Suslin constructed a counterexample to a claim by Lebesgue, who believed that the projection of any Borel set in $\mathbb{R}^{2}$ onto one of the coordinate axes is also a Borel set... | Solution. We need a so-called universal set $U$ in $\mathbb{R}^{3}$, which has two properties: it is closed itself, and any closed set in $\mathbb{R}^{2}$ can be obtained by intersecting this set with a plane orthogonal to the $O y$ axis (see problem II.2.30(b)). Note that by problem II.2.30, any Borel set $B$ in $\mat... | proof | Other | proof | Yes | Yes | olympiads | false | 33,659 |
32. Let sets $A_{1}, \ldots, A_{N} \subset\{1, \ldots, n\}$ be such that none of them is a subset of any other. Prove Sperner's lemma, which states that
$$
N \leqslant C_{n}^{[n / 2]}
$$
where $[x]$ is the integer part of $x \in \mathbb{R}$. | Solution. The number of all such sets of sets $C_{0}, C_{1}, \ldots, C_{n}$, such that
$$
\varnothing=C_{0} \subset C_{1} \subset \ldots \subset C_{n}=\{1, \ldots, n\}, \quad\left|C_{k}\right|=k
$$
is $n!$ ! The number of the specified sets containing $A_{i}$ is $\left|A_{i}\right|!\left(n-\left|A_{i}\right|\right)$ ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 33,660 |
33. Let $\mathscr{A}$ be an arbitrary system of subsets of $\Omega$. Prove that
$$
\sigma(\mathscr{A})=\bigcup_{\left\{A_{n}\right\}} \sigma\left(A_{1}, A_{2}, \ldots\right)
$$
where the union is taken over all at most countable subsystems $\left\{A_{n}\right\} \subseteq \mathscr{A}$.
In other words, no matter what ... | Solution. Obviously, $\sigma(\mathscr{A}) \supseteq \mathscr{B}$, where
$$
\mathscr{B}=\bigcup_{\left\{A_{n}\right\}} \sigma\left(A_{1}, A_{2}, \ldots\right)
$$
To prove the reverse inclusion, it is only necessary to establish that the system $\mathscr{B}$ is a $\sigma$-algebra. The system $\mathscr{B}$ is closed und... | proof | Other | proof | Yes | Yes | olympiads | false | 33,661 |
34. In metric spaces $(E, \rho)$ with metric $\rho$, the Borel $\sigma$-algebra $\mathscr{E}$ is defined as the system generated by open sets.
Show that in separable spaces $\mathscr{E}$ is generated by open balls. Verify that this is not the case in non-separable spaces. | Solution. If $X=\left\{x_{n}\right\}_{n \geqslant 1}$ is a countable dense system, then any open set $G$ in ( $E, \rho$ ) can be represented as
$$
G=\bigcup_{q, n} B_{q}\left(x_{n}\right), \quad B_{q}\left(x_{n}\right)=\left\{x: \rho\left(x, x_{n}\right)0$ the condition
$$
x, y \in D \Rightarrow \rho(x, y)>\varepsilo... | proof | Other | proof | Yes | Yes | olympiads | false | 33,662 |
35. Let $B$ be a Borel set on the real line with Lebesgue measure $\lambda$. We define the density of the set $B$ as the limit
$$
D(B)=\lim _{n \rightarrow \infty} \frac{\lambda(B \cap[-n, n])}{2 n}
$$
(assuming this limit exists).
(a) Provide an example of a set $B$ for which the density $D(B)$ is not defined (i.e.... | Solution. (a) For the Borel set
$$
B=\bigcup_{n=0}^{\infty}\left[2^{2 n}, 2^{2 n+1}\right]
$$
the density $D(B)$ is not defined. Indeed,
$$
\lim _{n} \frac{\lambda\left(B \cap\left[-2^{2 n}, 2^{2 n}\right]\right)}{2^{2 n+1}}<\frac{1}{4}<\lim _{n} \frac{\lambda\left(B \cap\left[-2^{2 n+1}, 2^{2 n+1}\right]\right)}{2^... | proof | Other | proof | Yes | Yes | olympiads | false | 33,663 |
36. (Completion of $\sigma$-algebras.) Let $(\Omega, \mathscr{F}, \mathrm{P})$ be a probability space. This space is said to be complete (or complete relative to the measure $\mathrm{P}$) if from $B \in \mathscr{F}$ and $\mathrm{P}(B)=0$, it follows that every subset $A \subseteq B$ belongs to $\mathscr{F}$.
Denote by... | Solution. (a) The class $\overline{\mathscr{F}}$ is closed under countable unions, since $\mathscr{N}$ and $\mathscr{F}$ are closed under countable unions. If $A \cup N \in \overline{\mathscr{F}}$, where $A \in \mathscr{F}$ and $N$ is contained in some set $B$ from $\mathscr{F}, \mathrm{P}(B)=0$, then $\overline{A \cup... | proof | Other | proof | Yes | Yes | olympiads | false | 33,664 |
1. Let $F(x)=\mathrm{P}((-\infty, x])$. Show the validity of the following formulas:
$$
\begin{aligned}
\mathrm{P}((a, b]) & =F(b)-F(a), \quad \mathrm{P}((a, b))=F(b-)-F(a) \\
\mathrm{P}([a, b]) & =F(b)-F(a-), \quad \mathrm{P}([a, b))=F(b-)-F(a-) \\
\mathrm{P}(\{x\}) & =F(x)-F(x-),
\end{aligned}
$$
where $F(x-)=\lim ... | Solution. The considered equalities follow from the additivity property of the measure $\mathrm{P}$, as well as the following limit relations, valid as $\varepsilon \downarrow 0$:
$$
\begin{gathered}
\mathrm{P}((a, b-\varepsilon]) \uparrow \mathrm{P}((a, b)), \quad \mathrm{P}((a-\varepsilon, b]) \downarrow \mathrm{P}(... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,665 |
2. Let $\mathrm{P}_{n}$ be a probability measure on $\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}\right)\right)$, and $F_{n}=F_{n}\left(x_{1}, \ldots\right.$ $\left.\ldots, x_{n}\right)$ be the corresponding distribution function, i.e.,
$$
F_{n}\left(x_{1}, \ldots, x_{n}\right)=\mathrm{P}_{n}\left(\left(-\inf... | Solution. The desired relation holds for any finite measure $\mathrm{P}_{n}$ on $\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}\right)\right)$. This can easily be obtained by induction, noting that
$$
F_{n-1}\left(x_{1}, \ldots, x_{n-1}\right)=\Delta_{a_{n} b_{n}} F_{n}\left(x_{1}, \ldots, x_{n}\right)
$$
is t... | proof | Other | proof | Yes | Yes | olympiads | false | 33,666 |
3. Let $F=F\left(x_{1}, \ldots, x_{n}\right)$ be some distribution function on $\mathbb{R}^{n}$. Then on $\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}\right)\right)$ there exists and is unique a probability measure $P$ such that
$$
\mathrm{P}\left(\left(a_{1}, b_{1}\right] \times \ldots \times\left(a_{n}, b_{... | Solution. The desired statement is established as follows. First, $\mathrm{P}$ is defined on all elements of the algebra $\mathscr{A}$, consisting of all finite unions of sets of the form
$$
\left(a_{1}, b_{1}\right] \times \ldots \times\left(a_{n}, b_{n}\right]
$$
using the formula given in the condition. Then, the ... | proof | Other | proof | Yes | Yes | olympiads | false | 33,667 |
4. Show that the distribution function $F=F(x)$ on $\mathbb{R}$ has at most a countable number of discontinuity points. What can be said about the corresponding result for distribution functions in $\mathbb{R}^{n}$? | Solution. Suppose that $F(x-)<F(x)$ at a point $x \in \mathbb{R}$, i.e., $x$ is a discontinuity point of the function $F$. Then, to the point $x$, we can assign a rational number $q(x)$ from the interval $(F(x-), F(x))$. Different discontinuity points $x$ and $y$ will correspond to different rational numbers $q(x)$ and... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,668 |
5. Show that each of the functions
$G(x, y)=I(x+y \geqslant 0), \quad G(x, y)=[x+y]-$ the integer part of the sum $x+y$ is right-continuous and increasing in each variable, but is not a (generalized) distribution function in $\mathbb{R}^{2}$.
What necessary and sufficient condition must the function $F(x, y)$ satisfy... | Solution. The necessary and sufficient condition is that the function $F$ is right-continuous and the inequality
$$
F(a, b)+F(c, d)-F(a, d)-F(c, b) \geqslant 0
$$
holds for any $a \leqslant b$ and $c \leqslant d$ (this is nothing but the measure of the rectangle $(a, b] \times(c, d])$. For the given functions $G$, th... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,669 |
6. Let the Lebesgue-Stieltjes measure $\nu$ correspond to some continuous generalized distribution function on $\mathbb{R}$. Show that if the set $A$ is at most countable, then $\nu(A)=0$. | Solution. From the continuity of the generalized distribution function, it follows that $\nu(\{x\})=0$ for any $x$. Therefore, by countable additivity, $\nu(A)=0$ when $A$ is at most countable. | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,670 |
7. Let $(\Omega, \mathscr{F}, \mathrm{P})$ be some probability space. As shown in problem II.1.6, the measure P allows us to introduce a "distance" between sets from $\mathscr{F}$ using the formula
$$
d(A, B)=\mathrm{P}(A \triangle B)
$$
(a) Establish the following important topological property of the $\sigma$-algeb... | Solution. (a) Essentially, it is required to establish that for any arbitrarily small $\varepsilon>0$ and any $B \in \sigma(\mathscr{A})$, there exists an $A \in \mathscr{A}$ such that $\mathrm{P}(A \triangle B) \leqslant \varepsilon$.
We will use the principle of suitable sets. Let
$$
\mathscr{B}=\{B \in \sigma(\mat... | proof | Other | proof | Yes | Yes | olympiads | false | 33,671 |
8. In this problem, we propose to conduct a topological proof of the Carathéodory theorem for probability measures.
Thus, let there be some algebra $\mathscr{A}$ of subsets of $\Omega$ and a countably additive probability measure P defined on it. According to the conclusion of problem II.3.7, if the extension of the m... | Solution. (a) First, note that for any $A \in \mathscr{A}$, it is evident that the following equalities hold:
$$
\mathrm{P}(A)=\inf \mathrm{P}\left(\bigcup_{n}\left(A_{n} \cap A\right)\right)=\inf \mathrm{P}\left(\bigcup_{n} A_{n}\right)=\mathrm{P}^{*}(A)
$$
(The first equality follows from the countable additivity o... | proof | Other | proof | Yes | Yes | olympiads | false | 33,672 |
9. (Continuation of problem II.3.7.) Let $\mathscr{A}$ be some algebra of subsets of $\Omega$ and $\mathscr{F}=\sigma(\mathscr{A})$. Let $\mu$ be a $\sigma$-finite measure on $\mathscr{F}$. Show that
(a) the measure $\mu$ on $\mathscr{A}$ may not be $\sigma$-finite;
(b) if the measure $\mu$ on $\mathscr{A}$ is $\sigm... | Solution. (a) Consider the half-line $\mathbb{R}_{+}$ with the $\sigma$-algebra of Borel sets $\mathscr{B}\left(\mathbb{R}_{+}\right)$ and the Lebesgue measure $\lambda$. Define the system of sets
$$
\mathscr{A}=\left\{A \in \mathscr{B}\left(\mathbb{R}_{+}\right): A \cap\left[0,2^{n}\right)+2^{n}=A \cap\left[2^{n}, 2^... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,673 |
10. Provide a direct proof of the equalities
$$
\begin{gathered}
\frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} e^{-\frac{x^{2}}{2}} d x=1 \\
\sum_{n=k}^{\infty} C_{n-1}^{k-1} p^{k}(1-p)^{n-k}=1, \quad(p, k) \in(0,1] \times \mathbb{N}
\end{gathered}
$$
showing that the Gaussian and negative binomial distributions are indee... | Solution. To verify the first equality, note that
$$
\left(\int_{\mathbb{R}} e^{-\frac{x^{2}}{2}} d x\right)^{2}=\int_{\mathbb{R}^{2}} e^{-\frac{x^{2}+y^{2}}{2}} d x d y=\int_{0}^{2 \pi} d \varphi \int_{0}^{\infty} r e^{-\frac{r^{2}}{2}} d r=2 \pi
$$
where in the last equality we have switched to polar coordinates $(... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,674 |
11. (To the Carathéodory theorem. I.) Provide an example showing that if a measure $\mu$ on an algebra $\mathscr{A}$ is finitely additive but not countably additive, then it cannot, in general, be extended to a countably additive measure on $\sigma(\mathscr{A})$. | Solution. Let $\Omega=\mathbb{Q}$, and $\mathscr{A}$ be the algebra of all possible finite unions of half-intervals of the form $(a, b]=\{r \in \mathbb{Q}: a<r \leqslant b\}, a, b \in \mathbb{Q}$,
for which we define $\mu((a, b])=b-a$. The given measure $\mu$ is a finitely additive function on $\mathscr{A}$. However, $... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,675 |
12. (To the Carathéodory theorem. II.) Give an example of a finite countably additive function $\mu$ defined on some algebra $\mathscr{A}$ that does not extend to a countably additive function $\mu$ on $\sigma(\mathscr{A})$ (possibly taking infinite values). | Solution. Let $\mathscr{A}$ consist of sets of the form $A\left(n_{1}, \ldots, n_{k+1}\right)=\left(0,1 / n_{1}\right] \cup\left(1 /\left(n_{2}+1\right), 1 / n_{2}\right] \cup \ldots \cup\left(1 /\left(n_{k}+1\right), 1 / n_{k}\right]$
and
$$
B\left(n_{2}, \ldots, n_{k}\right)=\left(1 /\left(n_{2}+1\right), 1 / n_{2}... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,676 |
13. Let $\mathrm{P}$ be a probability measure on the $\sigma$-algebra $\mathscr{F}$ of subsets of a set $\Omega$. Let $C \subseteq \Omega$, but $C \notin \mathscr{F}$. Show that the measure $\mathrm{P}$ can be extended while preserving the property of countable additivity to the $\sigma$-algebra $\sigma(\mathscr{F} \cu... | Solution. According to problem II. 2.20, we have
$$
\sigma(\mathscr{F} \cup\{C\})=\{A C \cup B \bar{C}: A, B \in \mathscr{F}\}
$$
Then the required extension can be defined as follows:
$$
\mathrm{P}(A C \cup B \bar{C})=\mathrm{P}^{*}(A C)+\mathrm{P}_{*}(B \bar{C})
$$
where $\mathrm{P}^{*}(E)=\inf \{\mathrm{P}(D): E... | proof | Other | proof | Yes | Yes | olympiads | false | 33,677 |
14. Using Zorn's Lemma, equivalent to the Axiom of Choice:
«if on a set $X$ there is a partial order $\prec$ such that every chain, i.e., a set of pairwise comparable elements, has a maximal element, then there exists an element $x_{\max} \in X$ such that $x \prec x_{\max}$ for all $x \in X$», show that every finitely... | Solution. For any $A \notin \mathscr{A}$, extend $\mathrm{P}$ to the algebra generated by $\mathscr{A}$ and $A$ in the same way as in the solution to problem II.3.13. Define a partial order $\prec$ on all such extensions, considering $\mathrm{P}_{1} \prec \mathrm{P}_{2}$ if $\mathrm{P}_{i}$ is an extension of $\mathrm{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 33,678 |
15. Let $\mu$ and $\nu$ be two probability measures on the space $(E, \mathscr{E})$, where $\mathscr{E}$ is some $\sigma$-algebra of subsets of the set $E$. Suppose that for each $\delta>0$ there exists $E_{\delta} \in \mathscr{E}$ such that $\mu\left(E_{\delta}\right)<\delta$ and $\nu\left(\bar{E}_{\delta}\right)<\del... | Solution. We should take
$$
D=E \backslash \bigcap_{N \geqslant 1} \bigcup_{n \geqslant N} E_{2^{-n}}
$$ | proof | Other | proof | Yes | Yes | olympiads | false | 33,679 |
16. Let $\mu$ and $\nu$ be two probability measures on the space $(E, \mathscr{E})$, where $\mathscr{E}$ is some $\sigma$-algebra of subsets of the set $E$. Lebesgue proved that the following decomposition always holds:
$$
\mu(B)=\int_{B} f d \nu+\mu(B D), \quad B \in \mathscr{E},
$$
for some set $D, \nu(D)=0$, and a... | Solution. (a) First, note that if $\mathscr{E}$-measurable functions $f_{n}=f_{n}(x), n \geqslant 1$, satisfy condition $(* *)$, then $\sup _{n} f_{n}$ also satisfies condition $(* *)$:
$$
\begin{aligned}
& \int_{B} \sup _{n} f_{n} d \nu= \\
& =\sum_{m \geqslant 1} \int_{B \cap\left\{\max _{i \leqslant m-1} f_{i}\lamb... | proof | Other | proof | Yes | Yes | olympiads | false | 33,680 |
17. Prove the following fundamental result about the structure of distribution functions: each such function $F=F(x)$ can be represented as
$$
F=\alpha_{1} F_{\mathrm{d}}+\alpha_{2} F_{\mathrm{abc}}+\alpha_{3} F_{\mathrm{sc}}
$$
where $\alpha_{i} \geqslant 0, \alpha_{1}+\alpha_{2}+\alpha_{3}=1$,
$F_{\mathrm{d}}$ - a... | Solution. The discrete component can be isolated by setting
$$
\alpha_{1}=\sum_{y}[F(y)-F(y-)], \quad F_{\mathrm{d}}(x)=\frac{1}{\alpha_{1}} \sum_{y: x \geqslant y}[F(y)-F(y-)],
$$
where the sum over $y$ is well-defined since the increasing function $F$ has at most a countable number of discontinuities. Clearly, $F-\... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,681 |
19. Let $X=\left(X_{t}\right)$ be a Borel function on $[0, T]$. Considering $t$ as time, $X$ as the observed process, and $[0, T]$ as the observation period for $X$, P. Lévy proposed the following formula for calculating the time $l_{X}(x, T)$ spent by the process $X$ at level $x \in \mathbb{R}$ over the time interval ... | Solution. The function
$$
F(x)=\int_{0}^{T} I\left(X_{t}<x\right) d t
$$
is monotone and has right limits. According to problem II.3.18, the derivative $F^{\prime}(x)$ exists (and is finite) almost everywhere (with respect to Lebesgue measure). Using the existence of the right limits $F(x+0), x \in \mathbb{R}$, it is... | proof | Calculus | proof | Yes | Yes | olympiads | false | 33,683 |
20. (a) Show that each number $x \in [0,1]$ admits a representation of the form
$$
x=\sum_{n=1}^{\infty} \frac{x_{n}}{3^{n}}
$$
where $x_{n} \in \{0,1,2\}, n \geqslant 1$.
(b) Show that if for $x$ there are two representations $x=\sum_{n \geqslant 1} \frac{x_{n}}{3^{n}}$ and $x=\sum_{n \geqslant 1} \frac{y_{n}}{3^{n... | Solution. (a) The desired decomposition can be constructed recursively. Specifically, let
$$
f(x)= \begin{cases}0, & 0 \leqslant x \leqslant 1 \\ 1, & 11
$$
Note also that for numbers $x$ with "terminating" representations, this procedure gives their "canonical" representation.
(b) Suppose there are two representati... | proof | Other | proof | Yes | Yes | olympiads | false | 33,684 |
21. Let $\mathscr{C}$ be the Cantor set on $[0,1]$.
(a) Show that the cardinality of the set $\mathscr{C}$ is the same as the cardinality of the set $[0,1]$.
(b) What are the sets
$$
\mathscr{C} \oplus \mathscr{C}=\{x+y: x, y \in \mathscr{C}\}, \quad \mathscr{C} \ominus \mathscr{C}=\{x-y: x, y \in \mathscr{C}\} ?
$$ | Solution. (a) By problem II.3.20(c), the cardinality of the set $\mathscr{C}$ is the same as that of the set of sequences consisting of 0 and 2. It is well known that the cardinality of the set $\{0,2\}^{\infty}$ is the continuum.
(b) We will show that $\mathscr{C} \oplus \mathscr{C}=[0,2]$. By definition, $\mathscr{C... | proof | Other | proof | Yes | Yes | olympiads | false | 33,685 |
22. Show that the Cantor set $\mathscr{C}$ is compact, perfect (i.e., closed and dense in itself, or in other words, without isolated points), and nowhere dense.
Remark. This is nothing more than a topological characterization of the Cantor set on the line. Specifically, any compact, perfect, nowhere dense set in $\ma... | Solution. Closedness and being dense in itself follow from problem II.3.20(d). Compactness follows from closedness and boundedness. Furthermore, if the Cantor set $\mathscr{C}$ were dense in some interval, then by the closedness of $\mathscr{C}$, the entire interval would belong to $\mathscr{C}$, and thus $\mathscr{C}$... | proof | Other | proof | Yes | Yes | olympiads | false | 33,686 |
23. Provide an example of two sets $A$ and $B$ on the real line $\mathbb{R}$, having zero Lebesgue measure, but such that $A \oplus B=\mathbb{R}$. | Solution. The desired example can be constructed by applying Problem II.3.21(b):
$$
A=\mathscr{C} \oplus \mathbb{Z}, \quad B=\mathscr{C},
$$
where $\mathscr{C}$ is the Cantor set on $[0,1]$. | proof | Other | math-word-problem | Yes | Yes | olympiads | false | 33,687 |
25. Provide an example of a $\sigma$-finite measure $\mu$ on $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$ which
(a) is not a Lebesgue-Stieltjes measure - in other words, for which there does not exist a non-decreasing right-continuous function $G=G(x)$ (generalized distribution function) such that $\mu((a, b]) = G(b) - G(a... | Solution. In both points, one can take the counting measure from Exercise II.6.68. | proof | Other | math-word-problem | Yes | Yes | olympiads | false | 33,689 |
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