TITLE: Meaning of measure zero
QUESTION [26 upvotes]: My book describes measure zero as following:

A set of points on the $x$-axis is said to have measure zero if the sum of the lengths of intervals enclosing all the points can be made arbitrarily small. If $f(x)$ is bounded in $[a,b]$, then a necessary and sufficient condition for the existence of $\int_a ^b f(x)dx$ is that the set of discontinuities have measure zero.

I don't really understand what is said here. How can the sum of the lengths of the intervals be made arbitrarily small? Wouldn't you have to move the points together? The only way I would see a set of points not obeying this is if it contains an infinite amount of points. Can someone clarify the meaning of 'measure zero' and its relation to the integral?

REPLY [3 votes]: Something that really clarified and made measure 0 makes sense to me was thinking about the rational numbers: $\mathbb{Q}$. The first important idea in understanding measure 0 is that you can measure an interval by its length meaning the measure of $[a,b]$ and $(a,b) = b-a$ (Assuming $b>a$). Measure can be extended to sets that aren't intervals but they are the easiest to start with. 
Now with this simple definition the measure of a point is easy since $x=[x,x]$ so measure of $x$ is $x-x=0$. Now since measure of a larger set can be thought of as the sum of intervals in the set this implies that the measure of the natural numbers, and integers are both 0 since the measure is 0 at every point. 
Now in classical analysis there are results about the size of infinities which I don't think I need to go into much depth in this answer but essentially an infinite set is countably large if you can create a set from the natural numbers that maps onto the other set, which essentially means for each element in that set you can assign a Natural number.
As a quick example the Integers are countable since you can match them up:
$1\to 0, 2\to1, 3\to-1, 4\to2$ etc.
These simple results, mappings between countable sets and the natural numbers being measure 0 can be used to prove that other sets that don't seem like they would be measure 0 are like $\mathbb{Q}$
Now the relation to the Riemann integral is that the Riemann integral can be pictured as sum of these infinitely many infinitesimally tiny rectangles. However to sum these rectangles you need some sort of distance measure and as I have just shown the measure of a point is 0. This means the integral of a point is 0 which is a classical result from calculus that $\int_a^af(x)=F(a)-F(a)=0$. 
Another classical fact about integrals is that they are linear which is the property that $\int_a^cf = \int_a^bf+\int_b^cf$ By this property we can see that if we are integrating a function f over the real line [0,2] but there is a discontinuity at 1 lets say that f(1) just doesn't exist for simplicity. We can use the linearity of the integral $\int_0^2 f = \int_0^1f+ \int_1^2f$. But this integral still doesn't work since it doesn't exist at 1 so we take out the 1 and take the limit which gives us $\int_0^2f = \lim_{\epsilon \to 0}\int_0^{1-\epsilon}f + \int_{1+\epsilon}^2f + \int_{1-\epsilon}^{1+\epsilon}f$. Since the third integral is 0 in the limit we can pretend that the discontinuity at 1 doesn't exist. And because of measure 0 of countable subsets of $\mathbb{R}$ we can actually do this same process with countably many points.