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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!

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    it makes me some noise that $n$ is continuous; unless I don't know a version of DCT for nets (not sequences) – 2017-02-13
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    Not sure if this is helpful, but if $\sup_{n} f(\cdot|n)$ is integrable, you would have the $h$ that you need. – 2017-02-13
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    Here an interesting version of DCT that could help http://math.stackexchange.com/a/1703428/406412 – 2017-02-13
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    here [1]: https://books.google.cl/books?id=4hIq6ExH7NoC&pg=PA415&lpg=PA415&dq=dominated%20convergence%20theorem%20with%20nets&source=bl&ots=patUSpd-Ox&sig=niWoLjdH3abvemF6im5GLv2c9wc&hl=en&sa=X&redir_esc=y#v=onepage&q=dominated%20convergence%20theorem%20with%20nets&f=false/%22this%20book%22 it says this theorem is false in general; sorry :( – 2017-02-13
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    hmmm, too bad, but thank you all for your help! Anybody else having an idea? – 2017-02-13

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