Simple question that relates Lebesgue measure and Lebesgue integral.

Question

Amid one of my studies, I came up with the following question (which I think mustn't be hard to answer):

Let's say we have some cube $Q \subset \Bbb R^n$ whose sides are parallel to the coordinate axis. I wonder if the equality below holds:

$$ \mu(Q) = \int_Q 1 \, dx. $$

Here, by $\mu$ I mean the Lebesgue measure of the cube $Q$ and the integral is also the usual Lebesgue integral. I am new to measure theory and thus I don't think I have (yet) the tools to go through this problem.

I'd love a suscint answer, but if someone is passionate enough to give a full justification I would even be more apreciatted.

Thanks for any help in advance.

Answer 1

It depends on how you defined the Lebesgue integral, but if you started with integrals of simple functions (as in linear combinations of indicator functions of measurable sets), then your equality is true by definition of the Lebesgue integral.