Supremum of measurable functions

Question

If $\{f_n\}$ is a sequence of measurable functions on the same measurable set then:

i) $\sup_{1\le i\le n} f_i$ is measurable for each $n$.

ii) $\sup f_n$ is measurable.

I don't quite follow the difference between these two statements.

Answer 1

As Nate Eldredge pointed out, the difference is that the first supremum is over a finite set while the second is over a countable set. The second statement can be deduced from the first one using monotonicity, but it is a priori stronger.

It is not a big problem if we know that $f\colon X\to\mathbb R$ is Borel measurable if and only if $\{x\mid f(x)\leqslant a\}$ is measurable for any $a\in\mathbb R$. Then $$\left\{x\mid \sup_nf_n(x)\leqslant a\right\}=\bigcap_n\{x\mid f_n(x)\leqslant a\}$$ and the RHS is a countable intersection of measurable sets, hence a measurable set.