Show that the limit superior and limit inferior of a sequence of measurable functions are also measurable

Asked Mar 26 '19 at 03:56

Active Mar 26 '19 at 04:44

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I need help in the following proposition:

Let $f_n$ be a sequence of lebesgue measurable functions $f_n : [a,b] \rightarrow \mathbb{\overline{R}}$ then lim sup $f_n$, lim inf $f_n$ $[a,b] \rightarrow \mathbb{\overline{R}}$ are both lebesgue measurable functions.

I don't know how to manipulate those functions to proove $ \{x\in[a,b]: f^*(x)> \alpha \} $ where $f^* $ is limit superior and $f_*$ is limit inferior

edited Mar 26 '19 at 04:44

asked Mar 26 '19 at 03:56

José Marín

2

Prove first that each $g_n(x) := \sup_{k\ge n}f_k(x)$ is measurable. – amsmath Mar 26 '19 at 04:02
Note that $\sup_k f_k(x) \le \alpha $ iff $f_k(x) \le \alpha$ for all $k$. – copper.hat Mar 26 '19 at 04:22

Show that the limit superior and limit inferior of a sequence of measurable functions are also measurable

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