# Egorov’s Theorem (?)

Let $$(X, \mathbb A, m)$$ be a measurable space and let $$\{f_n : X \to \mathbb R\}_{n \in \mathbb N}$$ be a sequence of Borel measurable functions. If such sequence converges $$m$$-almost everywhere to some Borel measurable $$f: X \to \mathbb R$$, I have to prove that for any $$\epsilon > 0$$ there exists $$A \in \mathbb A$$ with $$m(A) < \epsilon$$ and such that $$\sup_{x \in X \setminus A} |f_n(x) - f(x) | \to 0$$ as $$n \to + \infty$$.

My question is simple: Isn't this just Egorov’s Theorem?

Yes, it is and additionally you need a finite measure space. It is wrong for not-finite measures: For example take $$f_n = 1_{[n,n+1]}$$, then $$f_n \rightarrow 0$$ pointwise, but if $$\lambda(A) < \varepsilon < 1$$, then we must have $$\lambda([n,n+1] \setminus A ) >0$$. Thus $$\sup_{x \in \mathbb{R} \setminus A} |f(x) - 0| =1.$$