## Reverse Riesz Type Problem

Suppose and consider such that . Take a -finite measure space. Suppose is a measurable function such that for all .

Prove that .

**Solution:** Define the following functional . This is well defined and therefore, by the theorem of representation of linear functionals on we get that there exists a function such that . This means that we have $latex \displaystyle \int_X(f-h)gd\mu=0,\ \forall g \in L^q $. Now, take to be a decomposition of into countably many measurable spaces with finite measure. For we have that . Since the space has finite measure, all step functions are contained in , and this means that there exists a sequence of step functions such that , proving that and therefore almost everywhere on . Doing this for all we see that a.e. on and therefore .