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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

1.1

Scalar multiples are measurable. If $c > 0$ , then [given, L1] ${c f > a} = {f > a / c}$ ; 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 $- f$ .

givenL1

2.1

The threshold identities

step 1.1L1L2

${\max (f, g) > a} = {f > a} \cup {g > a}, {\min (f, g) > a} = {f > a} \cap {g > a}$

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]

3.1

Assume $f + g$ is pointwise defined. For every real $a$ ,

step 2.1L1

${f + g > a} = ⋃_{q \in Q} ({f > q} \cap {g > a - q}) .$

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]

4.1

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]

5.1

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]

6.1

Steps 1.1 through 5.1 prove all four claims.

step 1.1step 2.1step 3.1step 4.1step 5.1∎

Depends on

Used by

Every measurable function admits simple approximations dominated by its absolute value Corollary

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)