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Let $\phi, \psi : \mathbb{R} \rightarrow \mathbb{R}$ be step functions, with $\phi ≤ \psi$. Show that $\int{\phi} ≤ \int{\psi}$

I was thinking to convert it to a Riemann sum and show that the partitions of $\phi$ in [a,b] are less than $\psi$ in [c,d]?

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  • $\begingroup$ Is it not enough to construct two arbitrary step functions such that $\phi \leq \psi$ and integrate them over the same interval? $\endgroup$
    – mike van der naald
    Nov 17, 2016 at 2:31

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$\psi-\phi$ will be a step function which is non-negative. So just prove that any step function which is non-negative has a non-negative integral. That is obvious.

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