`Bandits.ArrayModel.hasLaw_fst_apply_arrayMeasure`🔗

Lemma

No docstring.

Types

𝓐 : Type u_1m𝓐 : MeasurableSpace 𝓐A measurable space is a space equipped with a σ-algebra.
𝓡 : Type u_2m𝓡 : MeasurableSpace 𝓡

Given

ν : ProbabilityTheory.Kernel 𝓐 𝓡A kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsMarkovKernel νA kernel is a Markov kernel if every measure in its image is a probability measure.
n : ℕ

Then

ProbabilityTheory.HasLaw (fun ω => ω.1 n) MeasureTheory.volume (arrayMeasure ν)

The predicate HasLaw X μ P registers the fact that the random variable X has law μ under the measure P, in other words that P.map X = μ.

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A kernel is a Markov kernel if every measure in its image is a probability measure.

ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)

A kernel from a measurable space `α` to another measurable space `β` is a measurable function
`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.

Nat : Type

The natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.

ProbabilityTheory.HasLaw : {Ω : Type u_1} →
  {𝓧 : Type u_2} →
    {mΩ : MeasurableSpace Ω} →
      {m𝓧 : MeasurableSpace 𝓧} →
        (Ω → 𝓧) → MeasureTheory.Measure 𝓧 → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasLaw._auto_1 → Prop

The predicate `HasLaw X μ P` registers the fact that the random variable `X` has law `μ` under
the measure `P`, in other words that `P.map X = μ`. We also require `X` to be `AEMeasurable`,
to allow for nice interactions with operations on the codomain of `X`. See for instance
`HasLaw.comp`, `IndepFun.hasLaw_mul` and `IndepFun.hasLaw_add`.

MeasureTheory.MeasureSpace.volume : {α : Type u_5} → [self : MeasureTheory.MeasureSpace α] → MeasureTheory.Measure α

`volume` is the canonical measure on `α`.

Bandits.ArrayModel.arrayMeasure : {𝓐 : Type u_1} →
  {𝓡 : Type u_2} →
    {m𝓐 : MeasurableSpace 𝓐} →
      {m𝓡 : MeasurableSpace 𝓡} → ProbabilityTheory.Kernel 𝓐 𝓡 → MeasureTheory.Measure (Bandits.ArrayModel.probSpace 𝓐 𝓡)

Probability measure for the array model of stochastic bandits.

Go to its page

Code

lemma hasLaw_fst_apply_arrayMeasure (ν : Kernel 𝓐 𝓡) [IsMarkovKernel ν] (n : ℕ) :
    HasLaw (fun ω : probSpace 𝓐 𝓡 ↦ ω.1 n) volume (arrayMeasure ν)

Proof

(hasLaw_eval_infinitePi (fun _ ↦ volume) n).comp (hasLaw_fst_prod _ _)

Meaning last changed in v4.35.0-rc2-1-g61e506b (2026-09-22), the 3th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 3 project declarations, 25 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Bandits.ArrayModel.hasLaw_fst_apply_arrayMeasure🔗

`Bandits.ArrayModel.hasLaw_fst_apply_arrayMeasure`🔗