Littlewood’s three principles (3)
by tzy9393
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
.