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