Is set $A=[0,1]^2\cap\mathbb{Q}^2$ Lebesgue measurable? I'm confused by the notion of measurable sets in measure theory.
If we take a set $A=[0,1]^2\cap\mathbb{Q}^2$, would it be Lebesgue measurable?
On one hand: if it is measurable, then outer measure $\lambda^*(A)=\lambda(A)$. And minimal rectange which covers set $A$ is $[0,1]^2$, it means $\lambda^*(A)=1$. So $\lambda(A)=1$.
But on the other hand: $A$ is just a countable union of measurable sets with measure $0$, so $\lambda(A)=0$
 A: It's not a single rectangle you are looking to cover the set with, it is a union of tiny rectangles.  Because the rational numbers are countable,  you can cover them with a countable collection of rectangles, the first of size $\frac \epsilon 2$, the second $\frac \epsilon 4$, etc.,  and that geometric series adds up to $\epsilon$.  So you can cover the rationals in that rectangle with as small a collection of open sets as you want, ergo it is size 0
A: There is a nice semi-rigorous, probabilistic intuition to answer such questions. The only assumption you need to make is that $Q$ is measurable, which I believe if you are asking the question that you are then you should already know that it is.
The intuition goes like this: If I pick a real number at random, what is the probability of picking a rational number? You should already know rigorously, and hopefully intuitively, that the answer is $0$. What if I want my rational number to also be between $0$ and $1$? Well, clearly whatever probability this event has, it should be smaller than picking any rational number at all, so this must also be $0$.
Now, what is the probability of independently picking TWO rational numbers between $0$ and $1$? Well, since the events are independent, it's just the product of the individual probabilities, so $0\times 0 = 0$.
This is an intuition that should convince you of an answer, so at least you know what direction to take when answering. Remember that measure theory and probability theory were in a sense designed to solve such problems at there core, and basic intuition is really helpful when you don't diverge too much from core problems.
