TITLE: What are some examples of proving that a thing exists by proving that the set of such things has positive measure?
QUESTION [42 upvotes]: Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in many parts of mathematics.

What is your favourite example of a proof of this type?

Here are some examples:

The probabilistic method in combinatorics As I understand it, a
typical pattern of argument is as follows.  We have a set $X$ and want to
show that at least one element of $X$ has property $P$.  We choose some
function $f: X \to \{0, 1, \ldots\}$ such that $f(x) = 0$ iff $x$ satisfies
$P$, and we choose a probability measure on $X$.  Then we show that
with respect to that measure, $\mathbb{E}(f) < 1$.  It follows that
$f^{-1}\{0\}$ has positive measure, and is therefore nonempty.

Real analysis One example is Banach's
proof
that any measurable function $f: \mathbb{R} \to \mathbb{R}$ satisfying
Cauchy's functional equation $f(x + y) = f(x) + f(y)$ is linear.
Sketch: it's enough to show that $f$ is continuous at $0$, since then
it follows from additivity that $f$ is continuous everywhere, which makes
it easy.  To show continuity at $0$, let $\varepsilon > 0$.  An
argument using Lusin's theorem shows that for all sufficiently small
$x$, the set $\{y: |f(x + y) - f(y)| < \varepsilon\}$ has positive
Lebesgue measure.  In particular, it's nonempty, and additivity then
gives $|f(x)| < \varepsilon$.
Another example is the existence of real numbers that are
normal (i.e. normal to every base).
It was shown that almost all real numbers have this property
well before any specific number was shown to be normal.

Set theory Here I take ultrafilters to be the notion of measure, an
ultrafilter on a set $X$ being a finitely additive $\{0, 1\}$-valued
probability measure defined on the full $\sigma$-algebra $P(X)$.  Some
existence proofs work by proving that the subset of elements with the
desired property has measure $1$ in the ultrafilter, and is therefore nonempty.
One example is a proof that for every measurable cardinal
$\kappa$, there exists some inaccessible cardinal strictly smaller than
it.  Sketch: take a $\kappa$-complete ultrafilter on $\kappa$. Make an inspired choice of function $\kappa \to \{\text{cardinals } <
\kappa \}$. Push the ultrafilter forwards along this function to give
an ultrafilter on $\{\text{cardinals } < \kappa\}$. Then prove that the set
of inaccessible cardinals $< \kappa$ belongs to that ultrafilter ("has
measure $1$") and conclude that, in particular, it's nonempty.
(Although it has a similar flavour, I would not include in this list the cardinal arithmetic proof of the
existence of transcendental real numbers, for two reasons.  First,
there's no measure in sight.  Second -- contrary to
popular belief -- this argument leads to an explicit construction
of a transcendental number, whereas the other arguments on this list
do not explicitly construct a thing with the desired properties.)


(Mathematicians being mathematicians, someone will probably observe that
any existence proof can be presented as a proof in which the set of things
with the required property has positive measure. Once you've got a thing
with the property, just take the Dirac delta on it.  But obviously I'm
after less trivial examples.)
PS I'm aware of the earlier question On proving that a certain set is
not empty by proving that it is actually
large. That has some good
answers, a couple of which could also be answers to my question. But my
question is specifically focused on positive measure, and excludes
things like the transcendental number argument or the Baire category
theorem discussed there.

REPLY [7 votes]: A very famous and important theorem in the theory of metric embeddings is known as "Assouad's Embedding Theorem". It concerns doubling metric spaces: metric spaces for which there is a constant $D$ such that every ball can be covered by $D$ balls of half the radius.

Theorem (Assouad, 1983): For every $\epsilon\in (0,1)$ and $D>0$, there are constants $L$ and $N$ such that if $(X,d)$ is doubling with constant $D$, then the metric space $(X,d^\epsilon)$ admits an $L$-bi-Lipschitz embedding into $\mathbb{R}^N$.

This theorem is widely used throughout metric geometry and analysis on metric spaces. (See, e.g., here or here.)
An $L$-bi-Lipschitz embedding is simply an embedding that preserves all distances up to factor $L$. It's easy to see that the doubling condition is necessary for this theorem to hold. Moreover, there are known doubling metric spaces (the Heisenberg group for one) that are doubling but do not admit a bi-Lipschitz embedding into any Euclidean space, so one cannot allow $\epsilon=1$ in Assouad's theorem.
This means, of course, that the constants $L$ and $N$ must blow up as $\epsilon\rightarrow 1$, and this is reflected Assouad's proof.
Except, that's not quite true. In a really surprising construction, Naor and Neiman showed in 2012 that the dimension $N$ in Assouad's theorem can be chosen independent of the ``snowflake'' parameter $\epsilon$ as $\epsilon\rightarrow 1$. (The distortion $L$ must necessarily blow up in general.) In other words, one need not use too many dimensions for the embedding, no matter how close $\epsilon$ gets to $1$. I believe this shocked many people.
The construction of Naor and Neiman is probabilistic: they construct a random Lipschitz map from $(X,d^\epsilon)$ into $\mathbb{R}^N$, and show that it is bi-Lipschitz with positive probability. The proof is also a nice application to geometry of the Lovasz Local Lemma.
Assouad's paper: http://www.numdam.org/article/BSMF_1983__111__429_0.pdf
Naor-Neiman's paper: https://www.cs.bgu.ac.il/~neimano/Naor-Neiman.pdf