Let $(X,\mathcal{B},\mu)$ be a measure space, and let $f:X\to[0,+\infty]$ be a simple function. Suppose that there are disjoint measurable sets $E_1,\ldots,E_m$ such that $f$ is supported on $E_1\cup\ldots\cup E_m$ and equals $c_i$ on each $E_i$ for some $c_i \in [0,+\infty]$. Show that $$ \mathrm{Simp}\int_{X} f \,\mathrm{d}\mu = \sum_{j=1}^{m}c_j\mu(E_j).$$

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An Introduction to Measure Theory

Exercise 1.4.32 (Integral of simple functions)

Let $(X, ℬ, μ)$ be a measure space, and let $f : X \to [0, + \infty]$ be a simple function. Suppose that there are disjoint measurable sets $E_{1}, \dots, E_{m}$ such that $f$ is supported on $E_{1} \cup \dots \cup E_{m}$ and equals $c_{i}$ on each $E_{i}$ for some $c_{i} \in [0, + \infty]$ . Show that

Simp \int_{X} f d μ = \sum_{j = 1}^{m} c_{j} μ (E_{j}) .

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Trivial.

eluthingol

2020-08-10 00:00

Exercise 1.4.32 (Integral of simple functions)

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