A step function can be defined to be a linear combination of a sequence of brick functions. My question is - Are step functions always Lebesgue integrable ?
2026-04-27 17:10:07.1777309807
Lebesgue integrability of step functions
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No: Consider the step function $$s(x) = \begin{cases} 0 & x = 0\\n^2 & {1\over n+1} < x \le {1\over n}, \ n \in \Bbb N_+\end{cases}$$ It is measurable, but not integrable on $[0,1]$.