Measure, Integration & Real Analysis · Problem 4B.10

Product Measures — Sheldon Axler

Mathematics — Measure Theory · density-one-zero-implies-empty-or-full

Problem

Suppose

E

is a Lebesgue measurable subset of

\mathbf{R}

such that the density of

E

equals

1

at every element of

E

and equals

0

at every element of

\mathbf{R} \setminus E

. Prove that

E = \emptyset

E = \mathbf{R}