Measure, Integration & Real Analysis · Problem 3A.2

Integration — Sheldon Axler

Mathematics — Measure Theory · dirac-measure-integral

Problem

Suppose XX is a set, S\mathcal{S} is a σ\sigma-algebra on XX, and c∈Xc \in X. Define the Dirac measure δc\delta_c on (X,S)(X, \mathcal{S}) by δc(E)={1if c∈E,0if c∉E.\delta_c(E) = \begin{cases} 1 & \text{if } c \in E, \\ 0 & \text{if } c \notin E. \end{cases} Prove that if f:X→[0,∞]f: X \to [0, \infty] is S\mathcal{S}-measurable, then ∫f dδc=f(c)\int f\, d\delta_c = f(c). [Careful: {c}\{c\} may not be in S\mathcal{S}.]

Browse the problem library