### Littlewood’s three principles (3)

Finally the principle states that every convergent sequence nearly converges uniformly. Let’s figure out what the word “nearly” means in this context and this brings us to the Egorov’s theorem.

Theorem 1(Egorov)Suppose is a sequence of measurable functions defined on a measurable set with , and assume almost everywhere on . Given , we can find a closed set such that and uniformly on .

*Proof:* We may assume for every since we can always find a subset which differs with by a measure zero set such that the convergence holds everywhere on , then we just rename by . Define

Note for fixed we have the relation and as . Then by the continuity of measures we know which implies we can find an integer such that . By construction, we have whenever and . Then we choose so that and let .

First we notice the relation . Next for any , we can choose such that when . Now we indeed have completed our job as for any we can find an integer such that for all we \left|f_{j}(x)-f(x)\right|<\delta<\frac{1}{n} for all .

Finally, as a technical requirement, we can find a closed subset such that which leads us to the desired property .