Sufficient conditions on functions to interchange limit and integral

Question

Dear Stack exchange Math crowd,

I have the following kind of limit:

$$ \lim_{n\to\overline{n}}{\int_x I(g(x,n)>\overline{g})f(x|n)dx} $$

with $n$ being continuous and not discrete, $I$ being the indicator function, $g$ being some smooth function, and $f$ being a smooth conditional probability density functions.

I know that by the Dominated convergence theorem I need two conditions for me being able to pull the limit inside the integral (and, ultimately, the functions):

1. Pointwise convergence - I get that by my smoothness assumptions on g() and f()

2. I need some function h(x) that dominates $I(g(x,n)>\overline{g})f(x|n)dx$ for all x and n. I am not 100% sure whether I get that just from $f$ being a smooth pdf and $g$ being the indicator function.

So my question is whether if I assume that $f$ and $g$ are smooth, I can conclude that the expression above is equal to

$$ {\int_x I(g(x,\overline{n})>\overline{g})f(x|\overline{n})dx} $$

?

You can probably tell I am not a mathematician, but a social scientist trying to build a simple illustrative model...

Your help would be very much appreciated!