Measure, Integration & Real Analysis · Problem 2E.6

Integration — Sheldon Axler

Mathematics — Measure Theory · egorov-type-theorem-infinity

Problem

Suppose

(X, \mathcal{S}, \mu)

is a measure space with

\mu(X) < \infty

. Suppose

f_1, f_2, \ldots

is a sequence of

\mathcal{S}

-measurable functions from

X

\mathbf{R}

such that

\lim_{k \to \infty} f_k(x) = \infty

for each

x \in X

. Prove that for every

\varepsilon > 0

, there exists a set

E \in \mathcal{S}

such that

\mu(X \setminus E) < \varepsilon

and

f_1, f_2, \ldots

converges uniformly to

\infty

E

(meaning that for every

t > 0

, there exists

n \in \mathbf{Z}^+

such that

f_k(x) > t

for all integers

k \geq n

and all

x \in E

). [The exercise above is an Egorov-type theorem for sequences of functions that converge pointwise to

\infty