Lebesgue's measure of simple sets in $\mathbb{R}^2$

Question

What's the Lebesgue measure of:

$A_1 = \left \{ (x, y) \in \mathbb{R}^2: 1 < x^2 + y^2 \leq 9 \right \}$

$A_2 = \left \{ (x, y) \in \mathbb{R}^2: x^2 + y^2 \leq 1 \right \}$

I clearly have problems with intuition what a Lebesgue measure is at all and reading the definition doesn't help me a lot, so if you could explain this clearly and possibly provide some materials to help me apprehend this concept, I'd really be thankful.

EDIT:

If Lebesgue measure coincides with the standard area, does that mean that the measure of $A_1$ is $8 \pi$ and the measure of $A_2$ is $\pi$?

Answer 1

The Lebesgue measure of a set is simply its volume (or in the two-dimensional case the surface area)

$A_2$ is a disk with radius $1$, so $\mathcal{L}(A_2)=\pi$

$A_1$ is a disk with radius $3$, but missing an inner centered disk of radius 1, so $\mathcal{L}(A_2)=9\pi-\pi =8 \pi$.

Your two cases were really easy, as one immediately sees the sets shapes. In more complicated cases, you need a more systematical way to calculate the Lebesgue measure. This is done by approximating your sets with boxes (their Lebesgue measure is trivial) and taking the limit of letting this boxes get smaller (results in better approximations).

This approximation can also be done in two consecutive steps (like it is done in the picture above). First taking boxes with fixed width and optimal height, and then later optimizing the width. (This is called two-dimensional integration).

Then $$\mathcal{L}^2(A_2) =\int_{A_2} d\mathcal{L}^2 = \int_{\{0\le x^2+y^2\le 1\}} d\mathcal{L}^2(x,y) \\ =\int_{-1}^1 d\mathcal{L}(x) \int_{-\sqrt{1-x^2}}^\sqrt{1-x^2} d\mathcal{L}(y)\\ =4 \int_0^1 \sqrt{1-x^2} \, d\mathcal{L}(x)=\pi$$

Answer 2

For simple sets, its just the area. So for $A_1$, it's $\pi (9-1) = 8\pi$. What about $A_2$?

As for reading materials, "baby Royden" isn't bad.

The main idea of Lesbegue measure is to take ordinary measurement (like from geometry class, or maybe calculus) and generalize it to work for a larger class of sets. In the real line, for instance, geometry class tells you how much length is in the interval $[1, 3]$ (2 units!), but not how large the set of irrational numbers between 1 and 3 is. Lesbegue assigns a number to this as well (it turns out also to be 2, and the set of rational numbers between 1 and 3 has size zero). And it deals with lots of other sets with "holes" in them, whether finitely many (not too tough to deal with) or infinitely many (harder, and that's why the definition is a couple of pages long in your book).

Once you get the definition in the line, though, the generalization to the plane isn't too bad -- sort of like extending one-variable riemann integration to two variables, only not quite as tricky.