Exercise 1.6 (Sums as integrals)

Question

Let $\left(\Omega, (p_\omega)_{\omega \in \Omega}ight)$ be a discrete probability space with the associated measure $\mu$, and let $f:\Omega	o [0,+\infty]$ be a function. Show that
$$\int_\Omega f \,\mathrm{d}\mu = \sum_{\omega \in \Omega} f(\omega) \cdot p_\omega$$

Elu Thingol · Accepted Answer

See the equivalent \href{../an_introduction_to_measure_theory/exercise_1-4-39}{Exercise 1.4.39} from \href{..}{Terence Tao}'s \href{../an_introduction_to_measure_theory}{An Introduction to Measure Theory}.

Exercise 1.6 (Sums as integrals)

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Exercise 1.6 (Sums as integrals)

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