Is $f$ Borel and/or Lebesgue measurable?

Question

The function $f: \mathbb R \rightarrow \mathbb R$ is defined by $$\begin{equation} f(x)=\begin{cases} 2^n, & x=n \in \mathbb Z \\ 0, & \text{otherwise} \end{cases} \end{equation}$$

Is $f$ Borel measurable? Is $f$ Lebesgue measurable?

Please help on this. I want to understand why... Trying to self-teach this module.

Answer 1

The easiest way is to notice that $f$ differs from the zero function (which is measurable) on a null-set. As null-sets are always measurable, $f$ is measurable.

You can also do it explicitly:

• for $2^n\leq a$, $f^{-1}(a,\infty)=\{n,n+1,n+2,\ldots\}$.
• for $a=0$, $f^{-1}(a,\infty)=\mathbb Z$.
• for $a<0$, $f^{-1}(a,\infty)=\mathbb R$.