Precomposition Commutes with Pointwise Operations, and Sections of Functions Are Precompositions

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $V$ and $W$ be sets and $T:V\to W$ a function; for a function $f$ on $W$, $f\circ T$ is the composite of Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (ii), and for $A\subseteq W$, $T^{-1}(A)$ is the preimage of $A$, as in Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set. For functions on $V$ and on $W$, positive and negative parts and absolute values, indicator functions, sums and multiples in $[0,\infty]$, and dyadic approximations are those of Positive Part, Negative Part and Absolute Value of a Real-Valued Function (i), Positive Part, Negative Part and Absolute Value of a Real-Valued Function (ii), Positive Part, Negative Part and Absolute Value of a Real-Valued Function (iii), Indicator Function of a Subset (i), Sums, Multiples and Products of Functions into [0,∞] (i), Sums, Multiples and Products of Functions into [0,∞] (ii) and Dyadic Approximations of a Map into [0,∞] (i).

(i) Positive part, negative part and absolute value. For every function $f:W\to\mathbb{R}$,

$$(f\circ T)^{+}=f^{+}\circ T,\qquad(f\circ T)^{-}=f^{-}\circ T,\qquad|f\circ T|=|f|\circ T.$$

(ii) Sums, multiples, indicators and suprema in $[0,\infty]$. Let $f,g:W\to[0,\infty]$ be functions, $c\in[0,\infty]$ and $A\subseteq W$. Then

$$(f+g)\circ T=f\circ T+g\circ T,\qquad(cf)\circ T=c\,(f\circ T),\qquad\mathbf{1}_{A}\circ T=\mathbf{1}_{T^{-1}(A)}.$$

Let $(f_{m})_{m\in\mathbb{N}}$ be a sequence of functions $W\to[0,\infty]$ and $h:W\to[0,\infty]$ a function with $h(w)=\sup_{m}f_{m}(w)$ for every $w\in W$, the supremum as in Order and Suprema in the Extended Real Line (iii). Then

$$(h\circ T)(v)=\sup_{m}\,(f_{m}\circ T)(v)\qquad(v\in V).$$

(iii) Dyadic approximations. For every function $g:W\to[0,\infty]$ and every $m\in\mathbb{N}$,

$$s_{m}(g)\circ T=s_{m}(g\circ T).$$

(iv) Sections of functions. Let $Z$ be a set, $f:V\times W\to Z$ a function, $v\in V$ and $w\in W$, let $f_{v}$ and $f^{w}$ be the sections of $f$, and let $i_{v}:W\to V\times W$, $w'\mapsto(v,w')$, and $i^{w}:V\to V\times W$, $v'\mapsto(v',w)$, be the insertion functions. Then

$$f_{v}=f\circ i_{v},\qquad f^{w}=f\circ i^{w}.$$