Completeness of measure in convergence in measure.

Question

I know

1. if $\mu$ is complete and sequence of functions $f_k$ are measurable then from $f_k\rightarrow f$ a.e. we get that $f$ is measurable.
2. if sequence of measurable functions $f_k$ convergence to $f$ a.e. then $f_k\rightarrow f$ in measure.

Questions:

A) what does mean $f_k\rightarrow f$ a.e. here? Does it mean pointwise convergence a.e.?

B) (in 2) we say "…if sequence of measurable functions $f_k$ converges to $f$ a.e….". But we don't know is $f$ measurable because we don't know is $\mu$ complete. So $f_k$ can converge in measure to $f$ regardless of whether $\mu$ is complete or not. Yes?

EDIT: C) in the definition of convergence in measure there are written that $f_k$ and $f$ are measurable. So I can conclude that in (2) there are hidden condition that $\mu$ is complete. (It looks like I don't understand something that important). Is it true?

EDIT: I know that Lebesgue measure is complete. But I don't know does we talk in (2) about Lebesgue measure or about measure in general.

Answer 1

Let $(X,\mathcal{N},\mu)$ denote the underlying measure space.

(a) It means that there is a set $C\subset X$ with $\mu(X\setminus C)=0$ s.t. $f_n(x)\to f(x)$ for each $x\in C$.

(b) The actual result is

If a sequence of measurable functions $f_n$ converges a.e. to a measurable function $f$ and $\mu(X)<\infty$, then $f_n\to f$ in measure.