Prove $s\in L^1 and \int f = \int s$

Asked Feb 27 '18 at 16:33

Active Feb 27 '18 at 16:33

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Prove the following statements:

If $f$ is in $L^1$ and a function $s$ equals $f$ almost everywhere, then $s$ is in $L^1$ and $\int f = \int s$.
If $\phi(x)$ is a step function in $L^0$, then so is $|\phi(x)|$, and $|\int_{\mathbb{R}} \phi| \leq \int_{\mathbb{R}} |\phi|$.

I am having trouble with the first proof. I know that since $f\in L^1$ then $f$ can be written as $f=g-h$ for functions $g,h\in L^0$. This is where I get stuck.

The second proof I am unsure where to start at all.

asked Feb 27 '18 at 16:33

rover2

1,552

what do you mean by $L^0$? – Yanko Feb 27 '18 at 16:45
This may give you an idea for 2 https://math.stackexchange.com/questions/284322/the-triangle-inequality-for-integrals – Yanko Feb 27 '18 at 16:46

Prove $s\in L^1 and \int f = \int s$

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