Measure, Integration & Real Analysis · Problem 3A.4

Integration — Sheldon Axler

Mathematics — Measure Theory · lower-riemann-integral-zero-positive-function

Problem

Give an example of a Borel measurable function

f: [0,1] \to (0, \infty)

such that

L(f, [0,1]) = 0

. [Recall that

L(f, [0,1])

denotes the lower Riemann integral, which was defined in Section 1A. If

\lambda

is Lebesgue measure on

[0,1]

, then the previous exercise states that

\int f\, d\lambda > 0

for this function

f

, which is what we expect of a positive function. Thus even though both

L(f, [0,1])

and

\int f\, d\lambda

are defined by taking the supremum of approximations from below, Lebesgue measure captures the right behavior for this function

f

and the lower Riemann integral does not.]