Constant Functions are Measureable

Last updated Nov 1, 2022

Let $X$ be a Set and let $\mathcal{M}$ be any Sigma Algebra on $X$. Likewise, let $Y$ be a Set and let $\mathcal{N}$ be any Sigma Algebra on $Y$. Suppose $f: X \to Y$ is such that $\forall x \in X$ $f(x) = c$ for some $c \in Y$. Then $f$ is $(\mathcal{M}, \mathcal{N})$-measureable.

Suppose $S \in \mathcal{N}$. If $c \in S$ then every element of $X$ maps into
$S$ and $f^{-1}(S) = X$. If $c \not\in S$, then no element of $X$ maps into $S$ and $f^{-1}(S) = \emptyset$. Since ${\emptyset, X} \subset \mathcal{M}$ by definition of Sigma Algebra, we see that $f$ is $(\mathcal{M}, \mathcal{N})$-measureable. $\blacksquare$