The Strict Level Sets of a Sequence of Nonnegative Measurable Maps above a Simple Function Form a Sequence of Measurable Sets

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space, let $\mathcal{S}=\mathcal{S}(X,\mathcal{F})$ be the set of simple functions of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (ii) (only these two clauses of that definition are used, and they involve no measure, so none is fixed here), and let $\mathcal{M}^{+}=\mathcal{M}^{+}(X,\mathcal{F})$ be the set of measurable maps into $[0,\infty]$ of Nonnegative Measurable Maps and Their Lebesgue Integral (i). A real number is an element of $[-\infty,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii), and $<$ is as in The Extended Real Line, Its Order, and [0,∞] (iii). Let $t\in\mathcal{S}$, and let $(f_{m})$ be a sequence in $\mathcal{M}^{+}$.

(i) There is exactly one sequence $(E_{m})$ in $\mathcal{F}$ with $E_{m}=\{x\in X:t(x)<f_{m}(x)\}$ for every $m\in\mathbb{N}$.