# Convergence in Measure, Different Definitions

Let $(X, \mu)$ be a measure space, $E \subseteq X$ measurable, and $f_n$ a sequence of measurable functions on $E$. If $f$ is another function on $E$, I have seen two definitions for what it means for $f_n$ to converges to $f$ in measure:

(i) For any $\epsilon > 0$, there is an $N$ such that for all $n \geq N$, $\{ x \in E : |f_n(x) - f(x)| \geq \epsilon \}$ has measure $< \epsilon$.

(ii) For any $\epsilon > 0$, there is a measurable set $A$ with measure $< \epsilon$, and an $N$ such that for all $n \geq N$ and all $x \in E \setminus A$, $|f_n(x) - f(x)| < \epsilon$.

Both these definitions are given in Royden ( (i) is given specifically for the Lebesgue measure, (ii) for general spaces). A few questions:

1 . It does not appear anywhere in the definition that $f$ is measurable. A pointwise limit of measurable functions is measurable provided $\mu$ is complete, but even for a complete measure I see no reason why $f$ should be measurable in the definition above unless we define convergence in measure to require this. I don't even see why $\{x \in E: |f_n(x) - f(x)| \geq \epsilon \}$ should be measurable.

2 . The same question for definition (ii).

3 . Whether the definitions imply each other. Definition (i) seems to imply definition (ii).

It is my understanding that $f$ needs to be measurable before talking about convergence in measure (for either definition). For question 3, you are correct that (i) implies (ii) by choosing $A = \{|f_n - f|>\epsilon\}$.
For the other direction, simply note that $E\setminus A \subset \{|f_n - f|<\epsilon\}$, so $A \supset \{ |f_n - f| > \epsilon \}$, and consequently $\mu[|f_n - f| > \epsilon ] \leq \mu(A) < \epsilon$.