I was working on an exercise related to the Monotone Convergence Theorem applied on Lebesgue integration, and one of its hypothesis is that, for each $n$, the function $f_n(x)$ has to be measurable. Below I write the full theorem:
Theorem. Let $\{f_k(x)\}$ be a nondecreasing sequence of nonnegative measurable functions with limit f. Then \begin{equation*} \int f d\mu =\lim \int fd\mu. \end{equation*}
My main problem on solving this exercises is on proving the measurability of $f_n(x)$, for each $n$. I think it might not be hard, but I am just not getting the point... Below I drop the example I was talking about in the beginning.
Example. Evaluate the following limit \begin{equation*} \lim_n \int_0^{n\pi} \cos\left(\frac{x}{2n}\right)x^2e^{-x^3}dx. \end{equation*} I defined my sequence $f_n$ as $f_n(x) =\cos\left(\frac{x}{2n}\right)x^2e^{-x^3}$, for $x \in [0,n\pi]$ and for each $n$. How would I prove that, for each $n$, $f_n(x)$ is a measurable function? I know the definition of a measurable function, which relates the inverse image being measurable for every open set on the domain, but I still don't get it!
Note I was able to prove all the other hypothesis and to solve the exercise, assuming that each $f_n$ was measurable.
Thanks for all the help in advance.
Your functions $f_n$ in the exercise are continuous. Continuous functions from $[a,b]$ to $\mathbb{R}$ are measurable (assuming the standard measures/topologies).
Edit: As Ilya pointed out, to apply the MCT we should rectify the issue that the functions $f_n$ are defined on different intervals $I_n$, as well as each integral. To do so, we can extend $f_n$ to $\tilde{f}_n \colon \mathbb{R}\to \mathbb{R}$ defined by $$\tilde{f}_n = f_n \chi_{[0,n\pi]}$$
Then by continuity, $f_n$ is measurable, and we know that $\chi_{[0,n\pi]}$ is measurable. Their product is thus measurable, and we find that $$\int_{\mathbb{R}}\tilde{f}_n = \int_0^{n\pi}f_n$$ allowing us to apply the theorem.