Measure, Integration & Real Analysis · Problem 5C.6

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · lebesgue-product-measure-completeness

Problem

Suppose

\lambda

denotes Lebesgue measure on

(\mathbf{R}, \mathcal{L})

, where

\mathcal{L}

is the

\sigma

-algebra of Lebesgue measurable subsets of

\mathbf{R}

. Show that there exist subsets

E

and

F

\mathbf{R}^2

such that •

F \in \mathcal{L} \otimes \mathcal{L}

and

(\lambda \times \lambda)(F) = 0

; •

E \subseteq F

but

E \notin \mathcal{L} \otimes \mathcal{L}

. [The measure space

(\mathbf{R}, \mathcal{L}, \lambda)

has the property that every subset of a measurable set with measure

0

is measurable. This exercise asks you to show that the measure space

(\mathbf{R}^2, \mathcal{L} \otimes \mathcal{L}, \lambda \times \lambda)

does not have this property.]