ConditionalProbability.lean`

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Project sources: Pinned project documentation, Polynomial Freiman-Ruzsa proof paper, arXiv v2

2 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.Probability.ConditionalProbability

Imported by 2 files

lemma ProbabilityTheory.cond_real_apply The axiomatic definition of conditional probability derived from a measure-theoretic one line 15 source
Source documentation
The axiomatic definition of conditional probability derived from a measure-theoretic one.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, Set, MeasurableSet, MeasureTheory.Measure, Eq, Real, MeasureTheory.Measure.real, ProbabilityTheory.cond, HMul.hMul, Inv.inv, Inter.inter
Proof uses: of_eq_true, ENNReal.toReal, ENNReal, Eq.trans, True, congrFun', congrArg, ProbabilityTheory.cond_apply, ENNReal.toReal_mul, ENNReal.toReal_inv, eq_self
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma cond_real_apply (hms : MeasurableSet s) (μ : Measure Ω) (t : Set Ω) :
    μ[|s].real t = (μ.real s)⁻¹ * μ.real (s ∩ t) := by
…
```
theorem ProbabilityTheory.cond_isProbabilityMeasure_of_real The conditional probability measure of any finite measure on any set of positive measure is a probability measure line 21 source
Source documentation
The conditional probability measure of any finite measure on any set of positive measure is a probability measure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, Set, Ne, Real, MeasureTheory.Measure.real, OfNat.ofNat, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.cond
Proof uses: ProbabilityTheory.cond_isProbabilityMeasure_of_finite, Eq, ENNReal, False.elim, False, Eq.mp, Eq.trans, congrFun', congrArg, ENNReal.toReal, Not, True, eq_self, not_true_eq_false, Top.top
8 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem cond_isProbabilityMeasure_of_real {α : Type*} {_ : MeasurableSpace α} {μ : Measure α}
    {s : Set α} (hcs : μ.real s ≠ 0) :
    IsProbabilityMeasure μ[|s] := by
…
```