LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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The L2 velocity sensor is well-defined

Statement

Let $(D, D, λ)$ , $H$ , and the finite-second-moment probability law $μ$ be as in the Definition. For every $ψ \in H$ , the integral defining $S_{ψ} (v)$ is finite and independent of the representatives of $v$ and $ψ$ . The map $S_{ψ} : H \to R$ is linear and continuous, with $∣ S_{ψ} (v) ∣ \leq ∥ v ∥_{H} ∥ ψ ∥_{H} .$ Its centered observable $f_{ψ} (v) = S_{ψ} (v) - \int_{H} S_{ψ} d μ$ is Borel measurable and belongs to $L^{2} (μ)$ .

For the general random-state interface, if $X$ is a supplied separable real Banach space and $U$ is strongly measurable with $E ∥ U ∥_{X}^{2} < \infty$ , its Bochner mean exists. For every finite family of bounded real-linear sensors $L_{j} : X \to R$ , $L_{j} (U)$ has a measurable square-integrable version and $E L_{j} (U) = L_{j} (E_{B} U) .$ For a family $L_{1}, \dots, L_{m}$ , define its covariance matrix $Σ \in R^{m \times m}$ by $Σ_{i j} : = E [(L_{i} (U) - E L_{i} (U)) (L_{j} (U) - E L_{j} (U))], 1 \leq i, j \leq m .$ This matrix is well-defined and, for every $c = (c_{1}, \dots, c_{m}) \in R^{m}$ , $c^{T} Σ c = E [{(\sum_{j = 1}^{m} c_{j} (L_{j} (U) - E L_{j} (U)))}^{2}] \geq 0.$ Scalar almost-everywhere values are interpreted through the measurable representatives furnished by strong simple approximation; if a pointwise scalar random variable is required, such a version is used.

Statement references

Finite-second-moment random velocity sensor law

The space $L^{p} (μ)$ as the quotient by null functions

The Lebesgue integral is linear on $L^{1} (μ)$

Holder's inequality for integrals, including the endpoint cases

Probability measures and probability spaces

Teschl, Topics in Real and Functional Analysis, Section 11.6

Tao, An Introduction to Measure Theory, Section 1.7

Bochner integrability criterion

Bounded linear maps commute with Bochner integration

Strongly measurable Banach-valued function

Facts & Assumptions

Given: A finite measure space $(D, D, λ)$ with $0 < λ (D) < \infty$ , the normed space $H = L^{2} (D, λ; R^{d})$ , a probability measure $μ$ with $\int_{H} ∥ v ∥_{H}^{2} d μ (v) < \infty$ , and $ψ \in H$ .

[A1]

These data and the definition of $S_{ψ}$ and $f_{ψ}$ are fixed by Finite-second-moment random velocity sensor law.

[F1]

An element of $H$ is an almost-everywhere equivalence class of measurable representatives (The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

Holder's inequality with conjugate exponents $2, 2$ makes products of two $L^{2}$ functions integrable and bounds their integral by the product of their norms (Holder's inequality for integrals, including the endpoint cases).

[F3]

The Lebesgue integral is linear on its $L^{1}$ space (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F4]

The measure $μ$ is a probability measure, so $\int_{H} 1 d μ = 1$ (Probability measures and probability spaces).

[F5]

A strongly measurable Banach-valued map with integrable norm is Bochner integrable, and bounded linear maps commute with its integral (Bochner integrability criterion, Bounded linear maps commute with Bochner integration). Strong simple approximation supplies the measurable versions used here (Strongly measurable Banach-valued function).

Proof

Given: The spatial measure space, finite-second-moment law, and weight in the Statement.

1.1A1F1F2givenalgebra

For measurable representatives $v$ and $ψ$ , the pointwise inequality $∣ v (x) \cdot ψ (x) ∣ \leq ∣ v (x) ∣ ∣ ψ (x) ∣$ and [F2] give $\int_{D} ∣ v \cdot ψ ∣ d λ \leq ∥ v ∥_{H} ∥ ψ ∥_{H} < \infty$ . Replacing either representative on a $λ$ -null set changes the product only on a null set, so the integral depends only on the classes [F1]. Thus $S_{ψ} (v)$ is finite and representative-independent.

2.1F3step 1.1algebra

By [F3], $S_{ψ}$ is linear. Applying the estimate in step 1.1 to $v - w$ gives $∣ S_{ψ} (v) - S_{ψ} (w) ∣ \leq ∥ v - w ∥_{H} ∥ ψ ∥_{H}$ , so $S_{ψ}$ is continuous and hence Borel measurable.

2.2A1F2F4step 1.1algebra

Squaring the estimate in step 1.1 and integrating over $μ$ gives $\int_{H} ∣ S_{ψ} (v) ∣^{2} d μ (v) \leq ∥ ψ ∥_{H}^{2} \int_{H} ∥ v ∥_{H}^{2} d μ (v) < \infty$ . Holder's inequality on the probability space $(H, B (H), μ)$ and [F4] then give $\int_{H} ∣ S_{ψ} ∣ d μ < \infty$ , so its centering integral exists. Subtracting this finite constant preserves Borel measurability and $L^{2} (μ)$ membership, proving the assertion about $f_{ψ}$ .

3.1F2F4F5givenalgebra∎

In the general Banach-state clause, Holder on the probability space gives $E ∥ U ∥_{X} \leq (E ∥ U ∥_{X}^{2})^{1 / 2} < \infty$ , so [F5] supplies the Bochner mean. Composing the strong simple approximants with each continuous $L_{j}$ gives scalar measurable approximants converging outside the same null set; taking zero on that set supplies a measurable version. The bound $∣ L_{j} (U) ∣ \leq ∥ L_{j} ∥ ∥ U ∥_{X}$ gives its finite second moment. Bounded-map interchange in [F5] proves the expectation identity. Center each of the finitely many sensor versions. Holder makes all products integrable, and for every finite real coefficient vector $z$ the covariance quadratic form equals $E [(\sum_{j} z_{j} (L_{j} (U) - E L_{j} (U)))^{2}] \geq 0$ , proving positive semidefiniteness without another choice principle.

The L2 velocity sensor is well-defined

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources