Measurable Function is Simple Function iff Finite Image Set/Corollary
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Corollary to Measurable Function is Simple Function iff Finite Image Set
Let $\struct {X, \Sigma}$ be a measurable space.
Let $f: X \to \R$ be a measurable function.
Then $f$ has a standard representation.
Proof
Applying the main theorem to a simple function yields the representation $(1)$:
- $\map f x = \ds \sum_{i \mathop = 1}^n y_j \map {\chi_{B_j} } x$
which is of the required form.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.7 \ \text{(iii)}$