Hello. I wasn't going to ask this question on here, since I'm sure I've been over-thinking and just need to take a minute to reevaluate my solution attempt, but I need to get this done ASAP.

Problem Statement: Let . Prove that if is measurable, then so is where and .

Attempt at solution: The first thing I proved is that the statement is true for open and closed cubes; this is straight forward. Then we assume is measurable, and so .

Now I'm not sure exactly what to do (in a succinct manner) to prove the result; I've basically scribbled down like 3 pages of scratchwork trying to make everything workout, but I keep getting the wrong inequalities, or I'm not quite sure what I'm doing is legit (very very new to measure theory here).

Obviously if the above is satisfied, then we also have and . This is clear from the definitions involved. So I guess this proves that is measurable. (Maybe this requires a more rigorous justification?).

From here, I need to prove that the equality holds, but I'm not really sure how to do this in a rigorous fashion (I've attempted a few time son scratchwork, but I can't get a succinct proof to fall out). I was trying to use the above epsilon-inequality, since I can work with the term since I have the cube-decomposition lemma for open sets in and the proof that the theorem holds for cubes, but I can't work with in any tangible way (or can I?).

By the way, it is correct to interpret as ?

Finally, here was my other attempt at the solution, which doesn't really seem rigorous:

. The first is of course with respect to coverings of per the definition of Lebesgue measure (its agreement with the outer measure), and the second is for coverings of just .

Any help would be appreciated,

Thanks!