I saw this post but was still a little confused. My book shows that to prove this it is sufficient to make use of the following identity:
$$\limsup_{i\rightarrow\infty}\phi_i(x) = \inf_i\{\sup_{j\geq i}\phi_j(x)\}$$
And I'm having a bit of trouble understanding what this says. I've gotten a better understanding of $\limsup$ from my post yesterday, but still a bit confused.
So for a sequence of functions $\phi_i$, the $\limsup$ for a given point x is the limit of the supremum of $\phi_i(x)$. This could be increasing as i increases, or decreasing, or staying the same. And this is the same as taking the infimum over i of the set of supremums of $\phi_j$ for $j\geq i$? This last part is confusing me.
So for all i, we look at the $\{\sup_{j\geq i}\phi_j(x)\}$. And then we look at the greatest lower bound of this set? How is this the same as working with the limit?
And finally, how does this help us make it measurable?
Thanks for any help!
Observe that when $k\geq j,$ $\sup_{i\geq k}\{\phi_{i}(x)\}\leq\sup_{i\geq j}\{\phi_{i}(x)\},$ since the set $\{\phi_{i}(x):i\geq j\}$ contains $\{\phi_{i}(x):i\geq k\}.$ Then the sequence $(\sup_{i\geq j}\{\phi_{i}(x)\})_{j\geq 1}$ is decreasing, so the limit of this sequence must equal its infimum, which proves the identity in your book.
Now from the answer you linked, $\overline{\phi}_{j}(x):=\sup_{i\geq j}\{\phi_{i}(x)\}$ is a measurable function for every $j\geq 1,$ and by our reasoning above, $(\overline{\phi}_{j})_{j\geq 1}$ is a decreasing sequence of functions. Therefore the pointwise limit is measurable, as we wanted to prove.