Understanding supremum / infimum of a sequence of functions in context of sequences of measurable functions

Question

I am learning Real Analysis from Stein & Shakarchi. I am at where they begin to describe properties of measurable functions. They are introducing some notation regarding supremums and infimums of sequences of functions that I don't understand.

What does $sup_n f_n(x)$ mean? In my mind, this is $f_n$ where n is the supremum of the set ${1, 2, 3, 4....}$, so in my mind $sup_n f_n = f(x)$?? But I don't believe this is correct. Similarly, in my mind, $\text{inf}_n f_n (x)$ means $f_1(x)$ since 1 is the infimum of the set $\{1, 2, 3....\}$. What is the correct intuition for the meaning of this?

Answer 1

It helps to be pedantic here:

• $\sup_n f_n(x)$ is not a function: it's a number. Specifically, it's the supremum of the set $\{f_1(x), f_2(x), \ldots\}$ which you should be familiar with from pre-measure theory real analysis.
• $\sup_n f_n$ is a function. Specifically, it's a function whose value at $x$ is $\sup_n f_n(x)$.

To be more clear, the author should have written "$\sup_n f_n$ is measurable".

You can define the other quantities analogously.

Hopefully this clears everything up for you.