Convergence in measurs of any measurable function.

Question

Can i say that any measurable function $f:\Bbb R^n\to \Bbb R$ there exist a sequence of simple measurable functions $\phi_n, \phi_n:\Bbb R^n\to\Bbb R$ converges to $f$ in measure? This is true in pointwise convergence as well known result. But i don't know about convergence in measurs. There is Lebsuge measure on $\Bbb R^n$.Thanks.

Answer 1

What do you mean by convergence in measure? So there is already a measure $\mu$ on the space $\mathbb{R}^n$, right?
So if $|\mu|< \infty$, the almost sure convergence clearly gives the convergence in law.
If $|\mu|=\infty$, well, you have to differentiate carefully between different notion of "convergence in measures" ( vague convergence, weak convergence, etc).
Also, you have to be careful if $\mu$ is $\sigma$ finite or not.