Measure, Integration & Real Analysis · Problem 1A.2

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integrable-indicator-function

Problem

Suppose

a \leq s < t \leq b

. Define

f : [a, b] \to \mathbf{R}

f(x) = \begin{cases} 1 & \text{if } s < x < t, \\ 0 & \text{otherwise.} \end{cases}

Prove that

f

is Riemann integrable on

[a, b]

and that

\int_a^b f = t - s