If a sequence of functions is zero almost everywhere and converges pointwise almost everywhere, does the same hold for a limit?

Question

Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions in $\mathcal{L}^p(\mathbb{R})$. Each $f_n$ is zero almost everywhere. Additionally, the sequence converges pointwise almost everywhere to some $f$.

Is $f$ equal to zero almost everywhere?

My problem is, that I don't se any relation between the sets of measure zero $N_n := \{x \in \mathbb{R} \colon f_n(x) \neq 0\}$ and the corresponding set $N$ of $f$.

How would I determine the pointwise limit of $f$?

Answer 1

Let $N_n$ be defined as above. The function may not converge on $N=\cup_{n \in \mathbb{N}} N_n$ and must converge everywhere else, and by countable additivity N has measure 0. So f is zero a.e.