Let f:[0,1] --> (0,$\displaystyle \infty $) be a (Lebesgue) integrable function

and define g to be: $\displaystyle g(x) = \sqrt{f(x)} $. Show that g is integrable.

Answer:

If g is a measurable function, I can easily show that it's integrable. It's my reasoning that g is measurable that may be suspect, I claim that:

Let O be an open set. We know that: $\displaystyle f^{-1}(O) = U$ is a measurable set.

Thus, $\displaystyle g^{-1}(O) = \sqrt{U}$, where $\displaystyle \sqrt{U}$ = {$\displaystyle \sqrt{x}: x \in U$}

So it remains to show that U measurable implies $\displaystyle \sqrt{U}$ is measurable, which i was able to show but is a bit long, so I'll skip it.

Does this seem right, though? Or did i make it way too complicated?