The measurability of $f(x) = \sum_{r_n \leq x} \frac{1}{2^n}$

Question

Let $\mathbb{Q} \cap [0,1] = \{ r_1, r_2, \ldots \}$ be an enumeration of the rationals and let $f : [0,1] \rightarrow \mathbb{R}$ defined by

$$ f(x) = \sum_{r_n \leq x} \dfrac{1}{2^n} $$

I need to show that $f$ is measurable. Maybe it is easy, but I don't have any idea to how do this. Any hint will be helpful.

Answer 1

$f$ is the pointwise limit of the sequence of functions:

$$f_N(x) = \sum_{n=1}^N \frac{1}{2^n} \chi_{[r_n,1]}(x).$$

The $f_N$ are measurable because they are simple functions. Do you know how to prove that the pointwise limit of a sequence of measurable functions is measurable?