Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions

Let $\mathbb{R}$ be the real numbers and let $(X,\mathcal{A})$ be a measurable space. A real-valued function on $X$ is called measurable when it is measurable with respect to $\mathcal{A}$ and the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$. Let $f$ and $g$ be measurable real-valued functions on $X$ and let $c$ be a real number. Then:

1. (Constants and indicators.) The constant function $x\mapsto c$ on $X$ is measurable, and for every $A\in\mathcal{A}$ the indicator function $\mathbf{1}_A$, equal to $1$ on $A$ and to $0$ off $A$, is measurable.

2. (Sums and scalar multiples.) The pointwise sum $f+g$ and the pointwise multiple $cf$ are measurable. Consequently, for every natural number $n$, all real numbers $c_1,\dots,c_n$, and all measurable real-valued functions $f_1,\dots,f_n$ on $X$, the linear combination $c_1f_1+\cdots+c_nf_n$ is measurable.

3. (Products.) The pointwise product $fg$ is measurable; consequently every finite product $f_1\cdots f_n$ of measurable real-valued functions on $X$ is measurable.

4. (Absolute value, maximum, and minimum.) The functions $|f|$, $\max(f,g)$, and $\min(f,g)$, formed pointwise on $X$, are measurable, where $|f|(x)=\max(f(x),-f(x))$ is the absolute value of $f(x)$.

5. (Pointwise limits.) Let $f_n$, indexed by the natural numbers $n$, be measurable real-valued functions on $X$ such that at every $x\in X$ the sequence $(f_n(x))_{n\in\mathbb{N}}$ converges to a real number, written $f(x)$. Then $f:X\to\mathbb{R}$ is measurable.