Hello,

If I have a function $\displaystyle X_n(x)$ where $\displaystyle x \in [0,1]$ and $\displaystyle X_n(x)$ is the n-binary digit expansion of $\displaystyle x$, how can I show that $\displaystyle X_n(x)$ is measurable?

The full question I'm looking at asks to show that

$\displaystyle \limsup_n_\rightarrow_\infty \frac{1}{n} \sum_{1}^{k} X_n(x)$ is measurable.

I know that when we have a sequence of measurable functions, their sum will be measurable, and dividing a measurable function by n will still leave it measurable as well. Finally, when dealing with a sequence of measurable functions than their $\displaystyle \limsup_n_\rightarrow_\infty$ will also be measurable.

I have most of the puzzle figured out, but have never dealt with n-binary digits before. If I can just show that this expansion is measurable, everything else will fall into place.

My textbook tells me that if a sequence of functions converges, then it's limit will be measurable. Doesn't that apply in this case, since the n-binary digit expansion is converging to some point in [0,1]?