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Definitions:

Simple Function: Any functions that can written in the form:$$s(x)=\sum_{k=1}^na_n\chi_{A_n}(x).$$ Note the finite terms here.

It should follow that neither all simple functions are step functions, nor all step functions are simple function. e.g. Would not Cantor Function or Devil's Staircase be example of step function but not simple (note again the finite)?

I am asking just to be clear because I read on online notes somewhere that all step functions are simple but not converse.

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A step function is a linear combination of charateristic functions of intervals. Since any interval is measurable, any step function is simple. On the other hand, the characteristic function of Cantor's set is simple, but not a step function.

Cantor's function is neither simple, nor a step function.

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  • $\begingroup$ simple depends on the measure right? like step functions are (Lebesgue measurable)-simple functions? $\endgroup$
    – BCLC
    Aug 20 at 1:20
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According to Wikipedia, a step function is a finite linear combination of indicator functions of intervals. So a step function must be simple since it takes finite values.

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  • $\begingroup$ simple depends on the measure right? like step functions are (Lebesgue measurable)-simple functions? $\endgroup$
    – BCLC
    Aug 20 at 1:20

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