Egorov's Theorem

• categories: Measure Theory, Theorem

Egorov’s theorem provides a relationship between pointwise convergence and uniform convergence for measurable functions, under the assumption of finite measure.

Statement:

Let $\{f_n\}$ be a sequence of measurable functions defined on a measurable set $E$ with finite measure, i.e., $\mu (E)<\infty$. If $f_n\to f$ pointwise almost everywhere on $E$, then for every $ϵ>0$, there exists a subset $A_{ϵ}\subseteq E$ such that:

• $\mu (A_{ϵ})<ϵ$.
• $f_n\to f$ uniformly on $E\setminus A_{ϵ}$.

Intuition:

While pointwise convergence can be “slow” and irregular, Egorov’s theorem guarantees that, outside of a small exceptional set of arbitrarily small measure, the convergence is uniform. This exceptional set can have arbitrarily small measure but may depend on $ϵ$.