Greetings! I’m excited to author my first post and correct an imbalance in the amount of analysis present in the posts on this blog. I will try to continue the tradition of the quality, depth, and exposition of the posts before me.
This will be the first of what I anticipate to be three posts that will introduce the basics of measure theory, which will go something like:
- This post; which will outline the problem of measure.
- A deeper discussion of (pre-/outer-)measures, an introduction to -algebras, a statement (and maybe proof of) Carathédory’s theorem, and an introduction to Lebesgue measure and a brief discussion of product measures.
- Constructing the Lebesgue integral.
After which I hope to be regularly posting on topics in functional analysis, ergodic theory, and (as I learn it next semester) harmonic analysis.
Analysis is the art of approximation, and no problem is more fundamental in geometry and analysis than assigning a size or a measure to a set in a meaningful way. When asked, one might be quick to answer that area of a circle with radius one is or the length of the interval is , but one would naturally struggle trying to calculate the area of this:
Since each of these sets are merely a collection of points in the plane, one may ask a deeper question: can we measure all subsets of the plane, or in space? This natural question has a surprising answer.
We’ll first consider the problem of measuring sets descriptively. What we’d like to construct is a function that takes a subset of our given set and produce a non-negative number (possibly , accounting for our inclination to assign infinite size to the entire real line). Let us consider for a moment what we’d expect from such a measure. Certainly the set containing no elements should receive zero measure. We might demand that if we take two completely disjoint sets and , that . This certainly agrees with our physical intuition. However, in the fashion of the Ancient Greeks, we’d like to measure sets exhaustively: taking a simple set (like the circle) and approximating its area with a sequence of (let’s say triangulated) polygons. Thus, we might require that if is a countable family of disjoint sets, that their union is measured similarly: .
In the special case of we might further request from our measure that it the unit interval receive measure (length) one, and that is invariant under translations for every . These requests from our measure are seemingly lax, surely we can find some measure that will assign every subset of a size. Yet even with this collection of geometrically appealing axioms, we find difficulty. Consider the following construction of Giuseppe Vitali.
Define an equivalence relation on by declaring that if and only if is a rational number. Let be a subset of that contains exactly one member from each equivalence class (I used the Axiom of Choice here, can you determine how?). Now for each , let
Think of as the following: we take and shift it to the right by units, and we shift the part that sticks our past precisely unit to the left. Then certainly . Moreover, one can show that the family partitions . Suppose that can be measured by our
magic measure . Then since is really just a shift modulo one, its measure is invariant. More precisely
for any . Moreover, since is countable, and is the disjoint union of the ‘s, we see then that
If , then we have that the right hand sum is which is not equal to one. However, if , then the right hand sum is zero, which is not one. So we have a contradiction. The only assumption we made was that was could be measured by , and so it follows that is not measurable. As an aside; although we haven’t constructed Lebesgue measure, this argument demonstrates that the axiom of choice implies the existence of a Lebesgue non-measurable set. One might ask if we need the axiom of choice to construct such a set, and the answer is yes! There is a model (The Solovay Model) in which every axiom of ZF holds, absent the axiom of choice, in which every subset of the reals is Lebesgue measurable.
This shows the existence of a non-measurable set given our intuitive geometric requests of our measure. In case you wondered if it gets better in larger dimensions, I’ll refer you to check out a similar result known fondly as the Banach-Tarski Paradox.
So now we’ve come face to face with the problem of measure. To cope with this problem, we reject some of our previous hypothesis about what a meaningful measure might be, and we rightfully (for the time being) do, by ignoring the translation-invariance in the definition of our measure (we’ll return to this discussion) and we define a measure as follows.
Let be a set and let be a collection of subsets of . A function is called a measure provided that and if is a countable family of disjoint sets in , then .
To return to our original question – which sets are in fact measurable – will therefore depend on which measure we choose since our collection of measurable sets will be precisely the domain of our measure. For example, if we assign every set measure zero, then this is trivially verified to be a measure, and thus every set is measurable. But this seems rather silly since we’d like our measure (in the case of ; well for ) to reflect our geometric intuition (non-empty boxes getting positive measure, etc).
Constructing a meaningful measure on seems like an onerous task (and the first time one does, it is). But we might instead try to determine the minimum (loosely-speaking) amount of information we’d need to know to construct such a measure. Just in the sense that a continuous, real-valued function is determined by its values on a dense set, or that that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disc, we can completely determine the area (if it exists) of a set with a (relatively) minimal amount of information.
Next time, we’ll delve deeper into exactly what information one needs to generate a measure, and exactly what sets are measurable under such a measure.