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Borel representatives make the convolution integrand Borel measurable

Statement

Let $\tilde{f}, \tilde{g} : R^{n} \to C$ be Borel measurable functions. Then

H (x, y) : = \tilde{f} (x - y) \tilde{g} (y)

is Borel measurable on $R^{2 n}$ . In particular, for each fixed $x \in R^{n}$ , the section $y \mapsto H (x, y)$ is measurable.

Facts & Assumptions

Given: Borel measurable functions $\tilde{f}, \tilde{g}$ on $R^{n}$ .

[A1]

The functions $\tilde{f}$ and $\tilde{g}$ are Borel measurable by hypothesis.

[L2]

The Borel product on $R^{n} \times R^{n}$ is the Euclidean Borel sigma-algebra on $R^{2 n}$ (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}).

[L3]

Composition with Borel functions preserves measurability, and measurable arithmetic operations preserve measurability (Composition with a Borel measurable outer map preserves measurability, Arithmetic and lattice operations preserve measurability whenever they are defined).

[L4]

Sections of product-measurable functions are measurable (Every section of a product-measurable function is measurable).

Proof

technique · direct

1.1

The map $T : R^{2 n} \to R^{2 n}$ given by [L2, L3, given, construct] $T (x, y) : = (x - y, y)$ is continuous, hence Borel measurable. Since $(u, v) \mapsto \tilde{f} (u)$ and $(u, v) \mapsto \tilde{g} (v)$ are Borel measurable on $R^{2 n}$ by [L2] and [L3], the functions $(x, y) \mapsto \tilde{f} (x - y)$ and $(x, y) \mapsto \tilde{g} (y)$ are Borel measurable.

L2L3givenconstruct

2.1

Multiplication on $C$ is continuous, so [L3] makes [L3, L4, step 1.1]

H (x, y) = \tilde{f} (x - y) \tilde{g} (y)

Borel measurable on $R^{2 n}$ . Then [L4] gives measurability of each section $y \mapsto H (x, y)$ .

L3L4step 1.1∎

Depends on

Used by

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)