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.