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

[[concept]]

Topics

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Remark

simple functions can be written as a complex linear combination of finitely many indicator functions.

Proof

Suppose $\varphi :E\to \mathbb{C}$ is a simple function where the range $\varphi (E)=\{a_1,a_2,\dots ,a_n\}$. Define the sets $A_i={\varphi }^{-1}(\{a_i\})$ Note that each $A_i$ is measurable because they are intersections of the measurable sets where $\mathrm{Re}(\varphi )=\mathrm{Re}(a_i)$ and $\mathrm{Im}(\varphi )=\mathrm{Im}(a_i)$.

For all $i\ne j$ we have $A_i\cap A_j=\varnothing$ and ${\bigcup }_{i=1}^n{A}_i=E$. ie, the $A_i$ form a finite partition of the domain $E$. Thus, for all $x\in E$, we can write $\varphi (x)={\sum }_{i=1}^n{a}_i\cdot 1_{A_i}(x)$ Which is a finite complex linear combination of indicator functions. \tag*{$\blacksquare$}

References

References

See Also

Mentions

Mentions

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