The Sets Where a Function Is at Least 1/n Form a Sequence of Measurable Sets

This helper adopts Measurable Functions and the Lebesgue Integral: Notation; the measure $\mu$ of that setting plays no role. Let $(X,\mathcal{F})$ be a measurable space and let $f\in\mathcal{M}^{+}(X,\mathcal{F})$, a measurable function into $[0,\infty]$. For $n\in\mathbb{N}$, $n\in\mathbb{N}_{0}$ by Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (i), and $n$ read in $\mathbb{R}$ is a real number by Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima (v), $\mathbb{R}$ being an ordered field by The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum (i); $1\le n$ in $\mathbb{R}$ by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (ii) and $0<1$ by Rules of Arithmetic and Order in an Ordered Field (vii), so $n\neq0$, and its reciprocal $1/n=n^{-1}$ in $\mathbb{R}$ is a real number by Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima (ii). A real number lies in $[-\infty,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii), and $\le$ is the order of The Extended Real Line, Its Order, and [0,∞] (iii).

(i) Superlevel sets at $1/n$. There is a sequence $(A_{n})_{n\in\mathbb{N}}$ in $\mathcal{F}$ with

$$A_{n}=\{x\in X:1/n\le f(x)\}\qquad(n\in\mathbb{N}).$$