Closure properties of Lebesgue measurable functions

Measurability is preserved by absolute values, countable suprema, infima, and limits, finite sums and differences, extension to a product space, invertible linear changes of variable, and $(x,y)\mapsto f(x-y)$.Suggest a slogan

All functions below are Lebesgue measurable functions with values in $[-\infty,\infty]$ unless stated otherwise.

For a Lebesgue measurable set $E\subseteq\RR^d$, the characteristic function $\chi_E$ is measurable.
If $f_n\colon\RR^d\to[-\infty,\infty]$ are measurable for $n\geq1$, then $\abs{f_n}$, $\sup_n f_n$, $\inf_n f_n$, $\limsup_n f_n$, and $\liminf_n f_n$ are measurable, and so is $\lim_n f_n$ wherever the sequence converges everywhere.
If $f,g\colon\RR^d\to\RR$ are measurable and finite-valued, then $f+g$ and $f-g$ are measurable.
If $f\colon\RR^{d_1}\to[-\infty,\infty]$ is measurable, then $F\colon\RR^{d_1}\times\RR^{d_2}\to[-\infty,\infty]$, $F(x,y)\coloneqq f(x)$, is measurable.
If $f\colon\RR^d\to[-\infty,\infty]$ is measurable and $T\colon\RR^d\to\RR^d$ is an invertible linear map, then $f\circ T$ is measurable.
If $f\colon\RR^d\to[-\infty,\infty]$ is measurable, then $(x,y)\mapsto f(x-y)$ is measurable on $\RR^d\times\RR^d$.

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Outer measure and the construction