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 Jan20 accepted Integral of simple functions in standard and non-standard representation Jan20 comment Integral of simple functions in standard and non-standard representation Thank you; that's exactly what I was looking for. I tried to edit your post but I couldn't because I wanted to change just two characters. The limits of the last and the third from last summation might be wrong. Jan19 comment Integral of simple functions in standard and non-standard representation Yes, it means that it is a non-negative measurable function. Jan19 comment Integral of simple functions in standard and non-standard representation I would like to prove it without using the linearity of the integral, i.e. just by using the definition of the integral of simple functions (see the latest revision of my question). To say it another way, this question is more about sum and set manipulations than about measure theory. Jan19 awarded Editor Jan19 revised Integral of simple functions in standard and non-standard representation added 542 characters in body Jan19 comment Integral of simple functions in standard and non-standard representation We do not need to use the dominated convergence theorem, nor the linearity of the integral. The simple functions I am talking about take just a finite number of real values (i.e. their image is finite). Jan19 comment Integral of simple functions in standard and non-standard representation Yes, I meant what Thomas said. Jan19 asked Integral of simple functions in standard and non-standard representation Jan18 awarded Informed Jan17 awarded Supporter Jan17 accepted An inequality about sequences in a $\sigma$-algebra Jan17 asked An inequality about sequences in a $\sigma$-algebra Jan6 accepted Inequalities with $\sin(z)$, $z \in \mathbb{C}$ Jan6 comment Inequalities with $\sin(z)$, $z \in \mathbb{C}$ @cameron: I thought we were supposed to find something more explicit. Anyway I'll accept your answer, since you took the time to write it. Jan4 comment Inequalities with $\sin(z)$, $z \in \mathbb{C}$ @cameron: I have already tried that. I end up with $\sinh^2y\leq\cos^2x$. Jan4 asked Inequalities with $\sin(z)$, $z \in \mathbb{C}$ Jan1 awarded Scholar Jan1 accepted sequence of complex numbers Jan1 awarded Student