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Arithmetic and lattice operations preserve measurability whenever they are defined

Statement

Let $(X, A)$ be a measurable space and let $f, g : X \to \overline{R}$ be measurable. Then:

$c f$ is measurable for every real scalar $c$ ;
$\max (f, g)$ , $\min (f, g)$ , $∣ f ∣$ , $f^{+}$ , and $f^{-}$ are measurable;
if $f + g$ is pointwise defined, then $f + g$ is measurable;
with the convention of The convention $0 \cdot \infty = 0$ is used only for pointwise products of measurable functions, the pointwise product $f g$ is measurable.

Facts & Assumptions

Given: A measurable space $(X, A)$ and measurable functions $f, g : X \to \overline{R}$ .

[L1]

Extended-real measurability is equivalent to measurability of the threshold sets ${h > a}$ . (Threshold characterisations of real-valued and extended-real-valued measurability)

[L2]

The positive and negative parts are $h^{+} = \max (h, 0)$ and $h^{-} = \max (- h, 0)$ . (The positive and negative parts of a function)

[A1]

In this proof, the pointwise product uses the page convention $0 \cdot (+ \infty) = 0 \cdot (- \infty) = 0$ .

Proof

technique · direct

1.1givenL1

Scalar multiples are measurable. If $c > 0$ , then ${c f > a} = {f > a / c}$ ; if $c < 0$ , then ${c f > a} = {f < a / c}$ ; and if $c = 0$ , the function is constant. So [L1] gives measurability of $c f$ , and in particular of $- f$ .

2.1step 1.1L1L2

The threshold identities [step 1.1, L1, L2] ${\max (f, g) > a} = {f > a} \cup {g > a}, {\min (f, g) > a} = {f > a} \cap {g > a}$ show via [L1] that $\max (f, g)$ and $\min (f, g)$ are measurable. By [L2], this proves measurability of $f^{+}$ and $f^{-}$ ; replacing $g$ by $- f$ also gives $∣ f ∣ = \max (f, - f)$ . [step 1.1, L1, L2].

3.1step 2.1L1

Assume $f + g$ is pointwise defined. For every real $a$ , [step 2.1, L1] ${f + g > a} = ⋃_{q \in Q} ({f > q} \cap {g > a - q}) .$ The inclusion from right to left is immediate. For the converse, if $f (x) + g (x) > a$ then either $f (x) = + \infty$ , in which case any rational $q > a - g (x)$ works, or $f (x)$ is finite and one may choose a rational $q$ with $a - g (x) < q < f (x)$ . Thus [L1] gives measurability of $f + g$ . [step 2.1, L1].

4.1step 3.1L1A1

Suppose first that $u, v : X \to [0, + \infty]$ are nonnegative and measurable. [step 3.1, L1, A1] If $a < 0$ , then ${u v > a} = X$ . If $a \geq 0$ , then ${u v > a} = ⋃_{q \in Q, q > 0} ({u > q} \cap {v > a / q}) .$ Again the inclusion from right to left is immediate. For the converse, if $u (x) v (x) > a$ , choose a rational $q$ with $0 < q < u (x)$ and $a / q < v (x)$ ; this is possible because either $u (x)$ is finite positive and the rationals are dense, or $u (x) = + \infty$ , in which case any sufficiently large positive rational works. Hence nonnegative products are measurable by [L1]. [step 3.1, L1, A1].

5.1step 2.1step 3.1step 4.1L2A1

For general measurable $f$ and $g$ , step 2.1 gives measurable nonnegative [step 2.1, step 3.1, step 4.1, L2, A1] functions $f^{+}, f^{-}, g^{+}, g^{-}$ . By step 4.1 the four products $f^{+} g^{+}, f^{-} g^{-}, f^{+} g^{-}, f^{-} g^{+}$ are measurable. Put $h_{+} : = f^{+} g^{+} + f^{-} g^{-}, h_{-} : = f^{+} g^{-} + f^{-} g^{+} .$ At each point, at least one of $h_{+}$ and $h_{-}$ is zero, because at least one of $f^{+}, f^{-}$ and at least one of $g^{+}, g^{-}$ is zero. So the difference $h_{+} - h_{-}$ is pointwise defined without the forbidden $\infty - \infty$ form, and step 3.1 makes it measurable. By the usual sign decomposition, $h_{+} - h_{-} = f g$ , with the convention [A1] at the $0 \cdot \infty$ points. [step 2.1, step 3.1, step 4.1, L2, A1].

6.1step 1.1step 2.1step 3.1step 4.1step 5.1∎

Steps 1.1 through 5.1 prove all four claims. [step 1.1, step 2.1, step 3.1, step 4.1, step 5.1].

Depends on

Used by

Almost-sure convergence of an independent series is a zero-one event Corollary
Every measurable function admits simple approximations dominated by its absolute value Corollary
Positive, negative, and truncated Sobolev functions Corollary
Sobolev maxima and minima form a lattice Corollary
The graph of a measurable function Rⁿ to R is Lebesgue null Corollary
The critical Riesz potential can diverge and be essentially unbounded Counterexample
Almost-sure convergence of a random series Definition
Complex Lp classes and Euclidean test-function conventions Definition
Direct integral of a measurable Hilbert field Definition
Discrete martingale transform Definition
Hᵖ atoms with a prescribed moment order Definition
Measurable fields of von Neumann algebras and their direct integrals Definition
Partial sums, row sums and sample means Definition
Predictable quadratic variation in discrete time Definition
Rademacher functions on the unit interval Definition
The Littlewood-Paley square function Definition
Total row variance and the Lindeberg condition Definition
Zero truncation at a positive level Definition
A clipped affine function keeps its zero region Example
A deterministic integral construction of a Gaussian process Example
Extremal length of a rectangle and of a round annulus by hand Example
Matching C¹ pieces across a hyperplane have no jump derivative Example
Newtonian potential of radial compact data Example
Partial sums of independent centered variables are a martingale Example
Polya urn proportion martingale Example
Product martingale from independent mean one factors Example
Square of a martingale minus quadratic compensator Example
The positive-type Gaussian on the real line and its cyclic model Example
The square of the Volterra operator has zero trace Example
Borel change of variables from the compact-support formula and Radon uniqueness Lemma
Bounded compact data give an everywhere finite Newtonian potential Lemma
Complex Lq norm recovery from finite simple dual tests Lemma
Diagonal multipliers form a von Neumann algebra Lemma
Finite Rademacher blocks are equidistributed Lemma
Fourier pairing for a finite measure and Schwartz data Lemma
Largest-summand bound for independent symmetric variables Lemma
Levy maximal bound from uniform tail bounds Lemma
Measurable dense selections for fields of nonempty compact sets Lemma
Measurable Gram-Schmidt and constant-field trivializations on dimension strata Lemma
Monomial integrals on the sphere and orthonormality on the distinguished torus Lemma

…and 29 more results.

Dependency tree · two levels

7 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

Sheldon Axler, Measure, Integration and Real Analysis, Section 2B (standard reference, not scraped)
John K. Hunter, Measure Theory, Definition 3.3 (standard reference, not scraped)