In the following problem, $(X, \mathcal{M}, \mu)$ is a measure space, measurable means with respect to $\mathcal{M},$ and integrable means with respect to $\mu.$
Suppose $f$ and $g$ are nonnegative measurable functions on $X$ for which $f^2$ and $g^2$ are integrable over $X$ with respect to $\mu.$ Show that $f.g$ is also integrable over $X$ with respect to $\mu.$
My trial:
First, observe that $$ f.g = \frac{1}{2}[(f + g)^2 - f^2 - g^2]. $$ then the proof follows directly, but I am confused about how to show that $(f + g)^2$ is integrable as the given did say that $f$ and $g$ are integrable. could anyone help me in this step please?
For me it would be easier to use $$ \vert f\cdot g \vert \leq f^2+g^2.$$ Thus, we can dominate our function by an integrable one and therefore it is integrable.