If I have a general measure space $(X,\mu)$, and I have a sequence $f_n \to f$ in measure, and $\lvert f_n(x) \rvert \leq M$ for all $n \in \mathbb{N}$, and $x \in X$, is it true that we must also have $\lvert f(x) \rvert \leq M$ for all $x \in X$?
I believe this is true almost-everywhere: for any fixed $\varepsilon > 0$, $\mu(\{x : \lvert f\rvert > M + \varepsilon\}) \leq \mu(\{x: \lvert f_n \rvert > M\}) + \mu(\{x: \lvert f_n - f \rvert > \varepsilon\})$, and by convergence in measure of $f_n$ this means $\mu(\{x : \lvert f \rvert > M + \varepsilon\}) = 0$, so $\lvert f\rvert \leq M$ $\mu$-a.e.
But does this need hold everywhere?