Measure, Integration & Real Analysis · Problem 2E.8

Integration — Sheldon Axler

Mathematics — Measure Theory · uniform-convergence-weighted-measure

Problem

Suppose

\mu

is the measure on

(\mathbf{Z}^+, 2^{\mathbf{Z}^+})

defined by

\mu(E) = \sum_{n \in E} \frac{1}{2^n}.

Prove that for every

\varepsilon > 0

, there exists a set

E \subseteq \mathbf{Z}^+

with

\mu(\mathbf{Z}^+ \setminus E) < \varepsilon

such that

f_1, f_2, \ldots

converges uniformly on

E

for every sequence of functions

f_1, f_2, \ldots

from

\mathbf{Z}^+

\mathbf{R}

that converges pointwise on

\mathbf{Z}^+

. [This result does not follow from Egorov's Theorem because here we are asking for

E

to depend only on

\varepsilon

. In Egorov's Theorem,

E

depends on

\varepsilon

and on the sequence

f_1, f_2, \ldots