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Nonconstant affine ridge pullbacks preserve Lebesgue nullity

Statement

Assume Countable Choice. Let $d \geq 1$ , $w \in R^{d} ∖ {0}$ , $b \in R$ , and let $E \subseteq R$ have Lebesgue outer measure zero. Then

${x \in R^{d} : w \cdot x + b \in E}$

is Lebesgue measurable and has $d$ -dimensional Lebesgue measure zero. Consequently, if two Lebesgue measurable real functions on $R$ agree outside a null set, their compositions with the same nonconstant affine form $x \mapsto w \cdot x + b$ agree almost everywhere on $R^{d}$ .

Statement references

Tao, An Introduction to Measure Theory, product measures and linear changes of variables

Facts & Assumptions

Given: Countable Choice, $d \geq 1$ , a nonzero vector $w$ , a real $b$ , and a null set $E \subseteq R$ .

[F1]

Under Countable Choice, every null subset of $R$ is contained in a Borel null set: outer regularity supplies open covers of arbitrarily small measure, and their countable intersection is Borel; Borel sets are Lebesgue measurable and completeness makes the resulting outer-null set measurable and null (Assuming countable choice, the Lebesgue outer measure of an arbitrary subset of $R^{n}$ is the infimum of the measures of the open sets containing it, Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable, Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume).

[F2]

The product of two sigma-finite measures satisfies Tonelli's theorem for nonnegative measurable functions (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F3]

Product Lebesgue measure agrees with Euclidean Lebesgue measure on Borel subsets of a Euclidean product (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}).

[F4]

Invertible linear maps scale Lebesgue measure by the absolute determinant; in particular an orthogonal map preserves it (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not).

[F5]

Translation and nonzero scalar dilation preserve Lebesgue nullity (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not).

[F6]

Lebesgue measure is complete under Countable Choice, so every subset of a measurable null set is measurable and null (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume).

[F7]

Under Countable Choice every Borel subset of $R^{d}$ is Lebesgue measurable (Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

[F8]

Under Countable Choice, Lebesgue measure in every positive dimension is $σ$ -finite: $R^{m} = ⋃_{r = 1}^{\infty} [- r, r]^{m}$ , and each box has finite measure $(2 r)^{m}$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F9]

A countable union of measurable null sets is measurable and null, by the measure property in [F6] and countable subadditivity (Finite and countable subadditivity of measures).

[A1]

Countable Choice means every sequence of nonempty sets has a function choosing one element from each (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1givenF4construct

Put $u = w / ∥ w ∥_{2}$ . For $d \geq 2$ , start with $u$ and inspect the standard basis vectors in their finite order; for each $e_{i}$ , subtract its orthogonal projection onto the span of the vectors already retained, and append and normalize the residual exactly when it is nonzero. Since the standard basis spans $R^{d}$ , this finite Gram–Schmidt procedure produces an orthonormal basis whose first vector is $u$ . Let $Q$ be the associated orthogonal coordinate map. If $d = 1$ , take $Q x = u x$ . In either case the first coordinate of $Q x$ is $u \cdot x$ , and [F4] says $Q$ preserves Lebesgue measure.

2.1step 1.1F1F5A1choose

Fix an integer $R \geq 1$ . Since $Q$ is a finite matrix, choose $C_{R} < \infty$ greater than $R$ times the largest sum of absolute entries in any row; then $Q ([- R, R]^{d}) \subseteq [- C_{R}, C_{R}]^{d}$ . Since $E$ has outer measure zero, [F1] makes it Lebesgue measurable and null. Thus $E_{R} = ((E - b) / ∥ w ∥_{2}) \cap [- C_{R}, C_{R}]$ is a one-dimensional measurable null set by [F5]. By [F1], choose a Borel null set $B_{R} \supseteq E_{R}$ . If $d = 1$ , the image under $Q$ of the set in the Statement intersected with $[- R, R]$ is contained in $B_{R}$ . If $d \geq 2$ , that image is contained in $B_{R} \times [- C_{R}, C_{R}]^{d - 1}$ .

3.1step 2.1F2F3F4F7F8A1

If $d = 1$ , the bounded pullback image is contained in the Borel null set $B_{R}$ , and the inverse orthogonal map preserves nullity by [F4]. If $d \geq 2$ , the Borel set $B_{R} \times [- C_{R}, C_{R}]^{d - 1}$ is Lebesgue measurable by [F7]; both Lebesgue factors are $σ$ -finite by [F8], so Tonelli [F2] gives it product measure zero because its one-dimensional section factor $B_{R}$ has measure zero, and [F3] identifies that product measure with $λ_{d}$ . Applying the inverse orthogonal map preserves nullity by [F4]. In either case the ridge pullback inside $[- R, R]^{d}$ is contained in a Borel null set.

4.1step 3.1F6F9A1∎

The full pullback is the countable union of its intersections with $[- R, R]^{d}$ , $R \geq 1$ . Each intersection is a subset of the Borel null set from step 3.1, hence is measurable and null by [F6]. The full pullback is measurable by [F6] and null by countable subadditivity [F9]. If two functions differ only on $E$ , their compositions differ only on this pullback, proving the consequence.

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Sources

Terence Tao, An Introduction to Measure Theory, product-measure and change-of-variables results (standard reference, not scraped)
Leshno, Lin, Pinkus, and Schocken, Multilayer Feedforward Networks With a Non-Polynomial Activation Function Can Approximate Any Function, §6 (standard reference, not scraped)