Integration of step functions

Question

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwarz inequality before, but I've hit a roadblock because I've no idea whether or not I can conclude that I(phi^2) is less than or equal to (I(phi))^2, which I need in order to get the result Any tips?

Answer 1

If $\phi$ is a step function, i.e., $\phi= \Sigma_{i=1}^n c_i\chi_i$ , then

$\phi $ is Riemann integrable, so it is Lebesgue integrable, so that

$\phi \in L^1$, so Cauchy-Schwarz applies:

$$ \int (\phi) (\phi)= \int \phi^2 \leq \int \phi \int \phi= (\int \phi )^2 $$

If you have not seen Lebesgue integration, then you can use the fact that the space of Riemann-integrable functions are an inner-product space (see* for a caveat) with inner-product given as above, i.e., $<f,g>:= \int fg$, and then apply Cauchy-Schwarz.

• The problem with Riemann integrable functions as a normed space is that we may have $||f||:= (\int f^2)^{1/2}=0$ without t$f$ being $0$, but this is not a problem in your situation, and you may work with Riemann-integrable functions as an inner-product space and apply Cauchy-Schwarz.