Are the results of uniform convergence of sequences of functions still valid for an uncountable index set?

Question

Results like interchanging order of limit and integral. How to see/prove that they are?

For example, for $f:\Bbb R\to\Bbb R$ and fixed $x,y$:

$\displaystyle\int\lim_{h\to 0}f(x+hy)=\lim_{h\to 0}\int f(x+hy)$

After some googling, I found that we can not talk about uncountable sequences of functions since by definition sequences are over a countable index set. I realize we don't need such a notion to prove the above example, but I would like to know how to see how we generalize.

Thank you.

Answer 1

I think you can simply use this statement

Statement: If $f:\Bbb R\to\Bbb R$ is any function (you can also change the domain and range to any metric space). Then $\lim\limits_{h\to 0} f(h) = L$ if and only if for every sequence $h_n$ such that $h_n\to0$ as $n\to\infty,$ we have $\lim\limits_{n\to\infty} f(h_n) = L$.