The below question is from Durrett's book on Probability theory. I wanted to ask why we need $f$ to be a measurable function fo part (i) of the question. My proof is as follows:
Since $ U_i $ are i.i.d, so are their functions. Now, $ \mathbb{E}f(U_i) = I$, and $\mathbb{E}|f(U_i)| = \int_0^1 |f(x)|dx < \infty$ by assumption, we have by the weak law of large numbers that $I_n \rightarrow I$ in probability.
