File size: 1,568 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import Mathlib
import OAI.Probability.LogConcave.Model
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology
section LowerProof
open Matrix Topology TopologicalSpace ProbabilityTheory Classical WithLp
open scoped Matrix.Norms.Elementwise
open WithLp
open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal
open Matrix
open Polynomial
open scoped BigOperators
namespace LogConcaveSampling
theorem TVAtMost.map {E F : Type*} [MeasurableSpace E] [MeasurableSpace F]
{μ ν : Measure E} {ε : ℝ} (h : TVAtMost μ ν ε)
{f : E → F} (hf : Measurable f) : TVAtMost (μ.map f) (ν.map f) ε := by
intro s hs
simpa only [Measure.real, Measure.map_apply hf hs] using h (f ⁻¹' s) (hs.preimage hf)
end LogConcaveSampling
namespace LogConcaveSampling
lemma prob_prod_event_lower {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(μ : Measure X) (ν : Measure Y) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
{s : Set (X × Y)} (hs : MeasurableSet s) {a : ℝ}
(h : ∀ y, a ≤ μ.real ((fun x => (x,y)) ⁻¹' s)) : a ≤ (μ.prod ν).real s := by
have h' : ENNReal.ofReal a ≤ (μ.prod ν) s := by
rw [Measure.prod_apply_symm hs]
calc
_ = ∫⁻ _ : Y, ENNReal.ofReal a ∂ν := by simp
_ ≤ _ := lintegral_mono (fun y =>
(ENNReal.ofReal_le_iff_le_toReal (measure_ne_top μ _)).mpr (h y))
exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top (μ.prod ν) s)).mp h'
end LogConcaveSampling
end LowerProof
end
end
end
end OAI
|