note, video

• Motivation for Lebesgue measure (read Tao 7.1)
• What is the definition of outer-measure?
• Hmm, the outer-measure of a closed box? Why is it so complicated?
  • Interlude, the Banach-Tarski sphere.

Discussion Time:

• How to prove that $\{1,2,3\}$ has zero outer-measure?
• How to prove that $\Z$ has zero outer-measure? And $\Q$?
• Can you summarize some rules that allow you get to the above results quicker?
• Can you show that $\R$ has zero outer-measure in $\R^2$? Can you prove that in general, a lower dimensional 'manifold' in $\R^n$ has zero outer measure? Like the unit circle in $\R^2$?
• (hard) How does outer-measure behave under product? Does $m^*(A \times B) = m^*(A) \times m^*(B)$?

Welcome to this class. So, I assumed you all had taken math 104 or the equivalent of it, which covers sequence and limits, metric space topology (open sets, distance functions, compact sets etc), and also some Riemann integrals. Why do we want to take the second course in analysis?

Here is what this course is mainly about: Lebesgue measure theory and integration, then some multivariable calculus (will be useful for differentiable manifold) and Fourier analysis (what functions can be approximated by sums of sine and cosine?).

Why do we need to use Lebesgue integral, rather than Riemann integral? Why isn't knowing the length of an interval $[a,b]$ to be $b-a$ enough? Before we say Riemann integral is not good enough, let's first recall what it is good for. Piecewise continuous functions are Riemman integrable, in particular piecewise constant function are Riemann integrable. But it is very limited. In particular, pointwise limit of Riemann integrable function may not be Riemann integrable (though uniform limit preserves it).

The problem is that, there are sometimes subsets in $\R$, which are not interval, but infinite unions of it. Our dream is that, given any subset $A$ of $\R$, we can assign to it a number, the 'measure' of $A$, that 'reflect' its size. Now, that is too vague. What do we mean when we say reflect its size? It better have some nice properties, like

• monotonicity: if $A \In B$, then $m(A) \leq m(B)$.
• additivity: if $A = B \sqcup C$, $A$ is the disjoint union of $B$ and $C$, then $m(A) = m(B) + m(C)$.
• translation invariance: for any $x \in \R$, such that $m(x+E) = m(E)$.

It turns out, it is impossible to find such a measure function on all subset, but it is only possible to define it for a sufficiently nice subset, which we will call, 'measurable subset'. There are certain desirable properties we want to have on measurable sets (they form a sigma-algebra, and contains all open subsets).

Let's pause and recall the definition of Boolean and Sigma-algebra. In computer science, you may have heard about the opeartion (not, and, or), defined on a set, here we have exactly the same notion, but in different notation, 'not'='complement', 'and'='intersection', 'or'='union'.

Ex: show that, if $S = \{a,b,c\}$, then the power set $2^S$ (the set of all subsets of $S$) consist of 8 elements, corresponding to $2^3=8$. Work out how the correspondence go.

Then, what is the sigma algebra? Sigma algebra is a stronger requirement than the Boolean algebra, since it allows for 'countably' many operations taken at a time.

:?:Why the word 'countable' is important? Can you replace it by the more general word 'infinite'?

Our plan in the following, is to first define the notion of an outer-measure $m^*(A)$, that works for all subset $A \In \R^n$. It has many nice properties, which we will spend this lecture and next explore. But, it is not a measure, because it fails the additivity condition. Then, we introduce the notion of Lebesgue measurability.

First, we define the measure of an open 'box' in $\R^n$. Then, for any given subset $A$, we cover it by countably many open boxes, and get an approximation of the outer-measure, and we optimize over the covering.

Lemma 7.2.5 as discussion problem. The trick, is to given oneself an epsilon of room, and cut $\epsilon$ into countably many small pieces $\epsilon = \epsilon \sum_{n=1}^\infty 1/2^n$.

Now, here is a tricky pit-fall. Given the definition of an outer-measure, how to compute the outer-measure of an open box? Is it possible that we use some trickery, we can cover a box using smaller boxes with a smaller total volume?

Let's see a few examples.

• Outer measure of $\Q$ in $\R$?
• Outer measure of $\R$ in $\R^2$?