Let $f:[a,b]\to\mathbb{R}$ be bounded. Prove that
$$\left|\int_a^{b^-}f(x)dx\right| \leq \int_a^{b^-}|f(x)|dx.$$Give an example showing that an analogue nonequality does not valid for the lower integral.
P.S. $\int_a^{b^-}$ means the upper integral.
Actually I tried to use the definitions and I think the inf and sup in the definitions of integrals make the difference in this case however I did not get that.
Appreciate any comment or solution.
Best Answer
Example for the lower integral. Let $f(x)=0$ if $x =\frac p q$ with $(p,q)=1$ and $q$ even, $f(x)=-1$ if $x =\frac p q$ with $(p,q)=1$ and $q$ odd, $f(x)=1$ if $x$ is irrational. The RHS is $0$ whereas LHS is $1$.
For the upper integral use the fact that $|\sup_{[c,d]} f(x)| \leq \sup_{[c,d]} |f(x)|$.