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If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.

My thoughts: We know for any sequence $A_n$, $\inf A_n\le\lim\inf A_n\le\lim\sup A_n\le\sup A_n$. But how do we define this for function $f_k$? I think we'll just define it pointwise for each $x$ in the domain of $f_k$. The same inequality should also be true for sequence of functions.

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Yes, the definition of $\sup_k f_k$ and so on is pointwise.

To show that these are measurable, consider that for every $a \in \mathbb R$ $$\left\{ x \;\middle|\; \sup_{k \in \mathbb N} f_k(x) \leq a \right\} = \bigcap_{k \in \mathbb N} \{ x \;|\; f_k(x) \leq a \}$$ and use the measurability of $f_k$ for $k \in \mathbb N$.

Also, $\limsup_{k \rightarrow \infty} A_k = \inf_{n \in \mathbb N} \sup_{k \geq n} A_k$ and finally $\limsup A_k = \lim A_k$ if the limit exists.

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