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If $f \sim g$ and $g \sim h$, then $f \sim h$. |
If $a \sim_F b$ and $b \sim_F c$, then $a \sim_F c$. |
If $f(a) \sim b$ and $a \sim c$, and $f$ is asymptotically equivalent, then $f(c) \sim b$. |
If $f$ is asymptotically equivalent to the constant function $c$, then $f$ converges to $c$. |
If two functions are eventually equal, then they are asymptotically equivalent. |
If $f$ and $h$ are asymptotically equivalent and $f \leq g \leq h$, then $f$ and $g$ are asymptotically equivalent, and $g$ and $h$ are asymptotically equivalent. |
If two functions are asymptotically equivalent, then they eventually have the same sign. |
If two functions are asymptotically equivalent, then they eventually have the same sign, and eventually have the same sign when negative or positive. |
If $f$ and $g$ are asymptotically equivalent and $f$ tends to $c$, then $g$ tends to $c$. |
If two functions are asymptotically equivalent, then they have the same limit. |
If $g(x)$ is $o(f(x))$, then eventually $sgn(f(x) + g(x)) = sgn(f(x))$. |
If two measures agree on the sets $\{x < \cdot\}$ for all $x$, then they are equal. |
If $F$ is a monotone function that is right-continuous, then the Lebesgue measure of the interval $[a,b]$ is $F(b) - F(a)$. |
If $F$ is a continuous increasing function on the real line, then the Lebesgue measure of the interval $[a, b]$ is $F(b) - F(a)$. |
If $F$ is a nondecreasing function and is continuous from the right, then the Lebesgue measure of the interval $(a,b)$ is $F(b) - F(a)$. |
The sets of the interval measure $F$ are the same as the sets of the Borel measure. |
The space of an interval measure is the entire universe. |
If $F$ is a continuous, monotone function, then the Lebesgue measure of the interval $[a,b]$ is $F(b) - F(a)$. |
If $F$ is a monotone right-continuous function, then the interval measure $\mu$ defined by $\mu(a,b] = F(b) - F(a)$ is a $\sigma$-finite measure. |
If $P(x)$ holds almost everywhere on $S$, and $P(x)$ implies $Q(x)$ for all $x \in S$, then $Q(x)$ holds almost everywhere on $S$. |
If $S$ is a null set, then the Lebesgue integral of $f$ over $S$ is zero. |
The Lebesgue measure on the real line is the same as the Borel measure on the real line. |
The space of the Lebesgue measure on a set $S$ is $S$. |
The Lebesgue measure of a set is always a Lebesgue measurable set. |
The complement of a Lebesgue measurable set is Lebesgue measurable. |
If two functions agree on a set $S$, then they are Lebesgue measurable on $S$ if and only if they are Lebesgue measurable on $S$. |
The Lebesgue measure on $\mathbb{R}$ is the same as the Lebesgue measure on $\mathbb{R}$ restricted to $\mathbb{R}$. |
If $S$ and $T$ are measurable sets, then the integral of $f$ over $S \cap T$ is equal to the integral of $f$ over $T$ restricted to $S$. |
If $S$ is a subset of $T$ and both $S$ and $T$ are Lebesgue measurable, then the integral of $f$ over $S$ is equal to the integral of $f$ over $T$ restricted to $S$. |
If $f$ is a Lebesgue-integrable function on a set $S$, then the integral of $f$ over $S$ is equal to the integral of $f$ over the whole space, restricted to $S$. |
The Lebesgue integral of any function over the empty set is zero. |
The Lebesgue integral of a function over the empty set is zero. |
If $f$ is a Bochner integrable function on a measurable space $M$, then $f$ is also Bochner integrable on the restriction of $M$ to any measurable subset $\Omega$. |
If $f$ is a function defined on a Lebesgue measurable set $S$, then $f$ is Lebesgue integrable on $S$ if and only if the function $x \mapsto f(x)$ if $x \in S$ and $0$ otherwise is Lebesgue integrable on $\mathbb{R}^n$. |
If $f$ is an integrable function on a set $T$ and $g$ is a function on a subset $S$ of $T$ such that $g(x) \leq f(x)$ for almost all $x \in S$, then $\int_S g \leq \int_T f$. |
If $f$ is a Borel measurable function defined on a set $S$, then the function $g$ defined by $g(x) = f(x)$ if $x \in S$ and $g(x) = 0$ otherwise is also Borel measurable. |
If $f$ is a Borel measurable function on $\mathbb{R}^n$, then $f$ is a Borel measurable function on any Borel subset of $\mathbb{R}^n$. |
If $f$ is a Borel measurable function on a set $S$, then the function $g$ defined by $g(x) = f(x)$ if $x \in S$ and $g(x) = 0$ otherwise is a Borel measurable function on $\mathbb{R}^n$. |
If $S$ and $T$ are Lebesgue measurable sets with $S \subseteq T$, then a function $f$ is Lebesgue measurable on $S$ if and only if the function $x \mapsto f(x)$ if $x \in S$ and $0$ otherwise is Lebesgue measurable on $T$. |
A function $f$ is Lebesgue measurable if and only if for every $a \in \mathbb{R}$ and every $i \in \{1, \ldots, n\}$, the set $\{x \in S \mid f(x)_i < a\}$ is Lebesgue measurable. |
A function $f$ is Borel measurable if and only if for all $a \in \mathbb{R}$ and $i \in \{1, \ldots, n\}$, the set $\{x \in S: f(x)_i \geq a\}$ is measurable. |
A function $f$ is Borel measurable if and only if for all $a \in \mathbb{R}$ and $i \in \{1, \ldots, n\}$, the set $\{x \in S: f(x)_i > a\}$ is measurable. |
A function $f$ is Borel measurable if and only if the preimage of every halfspace is a Borel set. |
A function $f$ is Borel measurable if and only if the preimage of every open interval is a Lebesgue measurable set. |
A function $f$ is Borel measurable if and only if the preimage of every closed set is a Lebesgue measurable set. |
A function $f$ is Borel measurable if and only if the preimage of every closed interval is a Lebesgue measurable set. |
A function $f$ is Lebesgue measurable if and only if the preimage of every Borel set is a Lebesgue measurable set. |
If $f$ is a Lebesgue-measurable function and $T$ is a Borel set, then the preimage of $T$ under $f$ is a Lebesgue-measurable set. |
A function $f$ is Lebesgue measurable if and only if the preimage of every Borel set is a Lebesgue measurable set. |
If $f$ is a continuous function defined on a Lebesgue measurable set $S$, then $f$ is Lebesgue measurable on $S$. |
The identity function is Lebesgue measurable. |
The identity function is measurable with respect to the Lebesgue measure on $S$. |
The Lebesgue measure on the real line is the same as the interval measure. |
The Lebesgue measure on $\mathbb{R}^n$ is the product measure of the Lebesgue measure on $\mathbb{R}$ with itself $n$ times. |
If $f_1, \ldots, f_n$ are nonnegative measurable functions on $\mathbb{R}$, then $\int_{\mathbb{R}^n} f_1(x_1) \cdots f_n(x_n) \, dx_1 \cdots dx_n = \int_{\mathbb{R}} f_1(x_1) \, dx_1 \cdots \int_{\mathbb{R}} f_n(x_n) \, dx_n$. |
The Lebesgue measure of an interval $[l, u]$ is $u - l$. |
The Lebesgue measure of the interval $[l, u]$ is $u - l$ if $l \<le> u$ and $0$ otherwise. |
The Lebesgue measure of a box is the product of the lengths of its sides. |
For any $c \in \mathbb{R}^n$, the set $\{c\}$ has Lebesgue measure zero. |
The Lebesgue measure of the open interval $(l, u)$ is $u - l$. |
The Lebesgue measure of an open interval is its length. |
The Lebesgue measure of an interval is its length. |
The Lebesgue measure of a box is the product of the lengths of its sides. |
The Lebesgue measure of a box is the product of the lengths of its sides, if the box is nonempty, and zero otherwise. |
The Lebesgue measure of a box is the product of the lengths of its sides, if the box is non-empty, and zero otherwise. |
The Lebesgue measure of a singleton set is zero. |
The Lebesgue measure of a closed interval is finite. |
The Lebesgue measure of a box is finite. |
The Lebesgue measure of a ball in $\mathbb{R}^n$ is finite. |
The Lebesgue measure of a ball in $\mathbb{R}^n$ is finite. |
The closed and open boxes are measurable. |
The Lebesgue measure of an interval is its length. |
If $l$ and $u$ are vectors in $\mathbb{R}^n$ such that $l \leq u$, then the Lebesgue measure of the box $[l, u]$ is $\prod_{i=1}^n (u_i - l_i)$. |
The Lebesgue measure of a box is the product of the lengths of its sides if the box is nonempty, and zero otherwise. |
The Lebesgue measure of a box is the product of the lengths of the sides of the box, if the box is nonempty, and zero otherwise. |
The Lebesgue measure of a singleton set is zero. |
The Lebesgue measure on the real line is sigma-finite. |
The Lebesgue measure of the entire space is infinite. |
If $A$ is a countable set, then the Lebesgue measure of $A$ is zero. |
If $A$ is a countable set, then $A$ is a null set. |
If $A$ is a finite set, then $A$ is a null set. |
The set $N$ is a null set if and only if the set $N \cup \{a\}$ is a null set. |
If $a$ is in $S$ and $S$ is a Lebesgue measurable set, then the set $N \cup \{a\}$ is a null set in the Lebesgue measure on $S$ if and only if $N$ is a null set in the Lebesgue measure on $S$. |
The Lebesgue measure on $\mathbb{R}^n$ is not the counting measure on any set $A \subset \mathbb{R}^n$. |
If $S$ is an open set, $C$ is a closed set, and almost every point of $S$ is in $C$, then $x \in S$ implies $x \in C$. |
If $C$ is a closed set and almost every point of the real line is in $C$, then $x \in C$. |
If the Lebesgue measure of a box is the product of the lengths of its sides, then the measure is the Lebesgue measure. |
The Lebesgue measure on $\mathbb{R}^n$ is the pushforward of the Lebesgue measure on $\mathbb{R}$ under the map $T(x) = t + \sum_{j=1}^n c_j x_j$, where $c_j |
The Lebesgue measure on $\mathbb{R}^n$ is the same as the measure obtained by translating the Lebesgue measure on $\mathbb{R}^n$ by $t$ and then scaling it by $c$. |
The Lebesgue measure on the real line is the pushforward of the Lebesgue measure on the real line under the affine transformation $x \mapsto t + c x$. |
If $P$ is a Borel-measurable predicate, then the set of $x$ such that $P(x)$ holds is almost everywhere equal to the set of $x$ such that $P(t + cx)$ holds. |
If $f$ is a non-negative measurable function, then $\int f(x) dx = c \int f(t + cx) dx$ for any $c \neq 0$. |
If $f$ is integrable, then so is $f(t + cx)$ for any $c \neq 0$. |
If $f$ is a real-valued function defined on the real line, then $f$ is integrable if and only if $f(t + cx)$ is integrable for any $t, c \in \mathbb{R}$. |
If $f$ is a real-valued function, then $\int f(x) dx = c \int f(t + cx) dx$ for any constant $c \neq 0$. |
If $c$ is a vector of nonzero real numbers, then the Lebesgue measure of the set $T = \{t + \sum_{j \in \text{Basis}} c_j x_j \mid x \in \mathbb{R}^n\}$ is equal to the product of the absolute values of the $c_j$'s. |
The Lebesgue measure on the real line is the pushforward of the Lebesgue measure on the real line under the affine transformation $x \mapsto t + c x$. |
If $f$ is a measurable function on the real line, then $\int f(x) dx = c \int f(t + cx) dx$ for any $c \neq 0$. |
The function $x \mapsto x \cdot y$ is Lebesgue measurable for any $y \in \mathbb{R}$. |
The Lebesgue measure of an affine transformation of a set is equal to the absolute value of the determinant of the transformation times the Lebesgue measure of the set. |
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