The mixture at each input is the weighted sum of output measures

AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture_apply · theorem · Teaching coverage

Statement

Let I be a finite index type, A and B measurable spaces, w:I→[0,∞) fixed weights, and Kᵢ:A→𝓜(B) s-finite kernels. For every input a∈A, the output measure of finiteMixture w K is exactly the sum of the scaled measures wᵢKᵢ(a,·). Normalization of w is not assumed. This evaluation identity explains the construction rather than asserting a new convergence property.

All objects and hypotheses

• {ι α β : Type*}, [Fintype ι], [MeasurableSpace α], [MeasurableSpace β] from the local section.
• w : ι → ℝ≥0 and κ : ι → Kernel α β.
• [∀ i, IsSFiniteKernel (κ i)].
• a : α is an arbitrary input.
• No ∑ᵢwᵢ=1 hypothesis, no Markovness, and no target measure. The primary equality is equality of measures, not merely equality of densities on measurable sets.

Notation and interpretation

Measurable space and kernel

A measurable space is a set with a specified sigma-algebra of events. A kernel K from A to B assigns to each input a a measure K(a,·) on B, and a↦K(a,E) is measurable whenever E is measurable. A kernel need not assign probability measures unless Markovness is separately assumed.

\[K:A\to\mathcal M(B),\qquad a\mapsto K(a,E)\text{ measurable for every measurable }E\subseteq B.\]

Measure evolved by a kernel; invariance

Write μK for the measure obtained by starting with μ and applying K. Lean writes μ.bind K or K ∘ₘ μ. Invariance says that this evolved measure is exactly μ. If μ is a probability law this means a stationary initial distribution, but the definitions and these theorems also allow arbitrary measures.

\[(\mu K)(E)=\int_A K(a,E)\,\mu(da),\qquad \mu K=\mu.\]

Composition and powers

Lean's L ∘ₖ K applies K first and L second. The multiplicative identity is the identity kernel a↦δ_a, and powers are repeated composition. All integrals here are nonnegative extended-real (lower Lebesgue) integrals; no signed or Bochner integrability is silently assumed.

\[(L\circ_{\!k}K)(a,E)=\int L(b,E)\,K(a,db),\quad K^0(a,\cdot)=\delta_a,\quad K^{n+1}=K^n\circ_{\!k}K.\]

Markov and s-finite kernels

Markov means every output measure has total mass one. An s-finite kernel is a countable sum of finite kernels, where each finite kernel has a finite uniform bound on its total mass across inputs. Markovness supplies this s-finiteness automatically. S-finite is not the same hypothesis as sigma-finite, and is not merely a pointwise finiteness assertion.

\[K(a,B)=1\ \forall a\quad\text{(Markov)};\qquad K=\sum_{n=0}^{\infty}K_n,\quad \forall n\ \exists C_n<\infty\ \forall a,\ K_n(a,B)\le C_n\quad\text{(s-finite)}.\]

Nonnegative weights

ℝ≥0 consists of finite nonnegative real numbers; ℝ≥0∞ also allows infinity and is the codomain of measure values. The coercion wᵢ↦cᵢ puts finite weights into extended nonnegative arithmetic. Zero weights are permitted, including when component masses are infinite; this arithmetic uses 0·∞=0.

\[w_i\in[0,\infty),\qquad c_i=\iota(w_i)\in[0,\infty].\]

Measurable equivalence and pushforward

A measurable equivalence e is a bijection whose forward and inverse maps are both measurable. Its pushforward measure records the original mass of inverse images. This is transport of measures, not a density transformation with an implicit Jacobian.

\[(e_*\mu)(E)=\mu(e^{-1}(E)),\qquad (e^{-1})_*(e_*\mu)=\mu.\]

Mathematical proof

classical
rw [finiteMixture, sum_apply]

apply Finset.sum_congr rfl
intro i _
rw [Kernel.withDensity_apply _ measurable_const, MeasureTheory.withDensity_const]

theorem finiteMixture_apply (w : ι → ℝ≥0) (κ : ι → Kernel α β)
    [∀ i, IsSFiniteKernel (κ i)] (a : α) :
    finiteMixture w κ a = ∑ i, (w i : ℝ≥0∞) • κ i a

theorem finiteMixture_apply (w : ι → ℝ≥0) (κ : ι → Kernel α β)
    [∀ i, IsSFiniteKernel (κ i)] (a : α) :
    finiteMixture w κ a = ∑ i, (w i : ℝ≥0∞) • κ i a := by
  classical
  rw [finiteMixture, sum_apply]
  apply Finset.sum_congr rfl
  intro i _
  rw [Kernel.withDensity_apply _ measurable_const, MeasureTheory.withDensity_const]

/-- Fixed normalized nonnegative weights mix Markov kernels into a Markov kernel.
Zero weights are allowed. -/

Scope and omitted-condition boundaries

• Evaluation interface only: no probability normalization, common target, invariance, reversibility or mixing result.
• A finite family of arbitrary state spaces is allowed; no finite-state restriction is introduced.
• This proof establishes that constant densities avoid withDensity's nonmeasurable zero branch.
• No independently checked primary textbook anchor; exact proof sources are the local definition and Mathlib identities.