I Limit of limits of linear combinations of indicator functions

[Unconscious]

I have a sequence of functions $0\leq f_1\leq f_2\leq ... \leq f_n \leq ...$, each one defined in $\mathbb{R}^n$ with values in $\mathbb{R}$. I have also that $f_n\uparrow f$.
Every $f_i$ is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles indicator functions.

Is there a way to prove that also $f$ is the limit of "step" functions?

 

[PeroK]

You need double hashes for Latex:

https://www.physicsforums.com/help/latexhelp/

 

[Svein]

Is there a way to prove that also is the limit of "step" functions?

In what sense? You certainly will have \int \vert f-f_{n} \vert \rightarrow 0 but I think you will have problems with other metrics .

 

[Unconscious]

I mean pointwise convergence, optionally except on a measure-zero set of $\mathbb{R}^n$.