simple functions can be written as a finite complex linear combination of indicator functions

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Proof

Suppose $ϕ : E \to C$ is a simple function where the range $ϕ (E) = {a_{1}, a_{2}, \dots, a_{n}}$ . Define the sets

A_{i} = ϕ^{- 1} ({a_{i}})

Note that each $A_{i}$ is measurable because they are intersections of the measurable sets where $Re (ϕ) = Re (a_{i})$ and $Im (ϕ) = Im (a_{i})$ .

For all $i \neq j$ we have $A_{i} \cap A_{j} = \emptyset$ and $⋃_{i = 1}^{n} A_{i} = E$ . ie, the $A_{i}$ form a finite partition of the domain $E$ . Thus, for all $x \in E$ , we can write

ϕ (x) = \sum_{i = 1}^{n} a_{i} \cdot 1_{A_{i}} (x)

Which is a finite complex linear combination of indicator functions.

\begin{matrix} ◼ \end{matrix}

File	Last Modified
Functional Analysis Lecture 10	2025-07-14
simple functions are measurable	2025-07-14

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Last Modified

Functional Analysis Lecture 10

2025-07-14

simple functions are measurable

2025-07-14

simple functions can be written as a finite complex linear combination of indicator functions

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