LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A conditional mixture is a probability path

Statement

Let $λ$ , $π$ , and $ρ$ satisfy the conditional probability-density path definition, and for each $t \in [0, 1]$ set $p_{t} (x) = \int_{R^{d}} ρ_{t} (x ∣ x_{1}) π (d x_{1})$ . Then $p_{t}$ is a nonnegative measurable function and $\int_{R^{d}} p_{t} (x) λ (d x) = 1$ for every $t$ ; hence it is a probability density with respect to $λ$ .

Statement references

Lipman et al., §3.1

Holderrieth and Erives, §3.1

Facts & Assumptions

Given: A sigma-finite Borel measure $λ$ , a Borel probability measure $π$ , and a conditional density $ρ_{t}$ normalized for $π$ -almost every endpoint.

[F1]

Probability paths and velocity fields: $ρ_{t} (x ∣ x_{1})$ is product-measurable, nonnegative, and has $λ$ -integral $1$ for $π$ -almost every $x_{1}$ ; $p_{t} (x)$ is its $π$ -mixture.

[F2]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for a nonnegative product-measurable function on two sigma-finite measure spaces, the section integrals are measurable and either iterated integral equals the product integral.

Proof

Given: The hypotheses in the Statement.

1.1givenF1F2

Fix $t \in [0, 1]$ . The measures $λ$ and $π$ are sigma-finite, and $(x, x_{1}) \mapsto ρ_{t} (x ∣ x_{1})$ is nonnegative and product-measurable by [F1]. Tonelli's theorem [F2] therefore makes $x \mapsto p_{t} (x)$ measurable and gives $\int_{R^{d}} p_{t} (x) λ (d x) = \int_{R^{d}} (\int_{R^{d}} ρ_{t} (x ∣ x_{1}) λ (d x)) π (d x_{1}) .$

2.1step 1.1F1algebra∎

The inner integral in step 1.1 equals $1$ for $π$ -almost every $x_{1}$ by [F1]. Since $π$ is a probability measure, the right-hand side is $\int 1 d π = 1$ . The mixture is nonnegative by [F1], so $p_{t}$ is a probability density with respect to $λ$ .

Source notes

The normalization is proved locally by Tonelli; the source citations document the conditional-to-marginal mixture convention.

Depends on

Used by

Dependency tree · two levels

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Sources

Yaron Lipman et al., Flow Matching for Generative Modeling, §3.1 (standard reference, not scraped)
Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.1 (standard reference, not scraped)