Q:

What are some examples of proving that a thing exists by proving that the set of such things has positive measure? 

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. 

A:

Szemerédi's theorem asserts that every set $A$ of integers of positive upper density (thus $\limsup_{N \to \infty} \frac{|A \cap [-N,N]|}{|[-N,N]|} > 0$) contains arbitrarily long arithmetic progressions.   One of the shortest (but not the most elementary) proofs of this remarkably deep theorem deduces it from a result in ergodic theory:

Furstenberg recurrence theorem: Let $E$ be a subset of a probability space $(X,\mu)$ of positive measure, and let $T: X \to X$ be an invertible measure-preserving shift.   Then for any $k \geq 1$ there exists a positive integer $n$ such that $E \cap T^n E \cap T^{2n} E \cap \dots \cap T^{(k-1) n} E$ has positive measure. 

The case $k=1$ is trivial, and the case $k=2$ is the classical Poincare recurrence theorem.   The general case was established in
Furstenberg, Harry, Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, J. Anal.  Math.  31, 204-256 (1977).  ZBL0347.28016. 
Roughly speaking the deduction of Szemerédi's theorem from Furstenberg's theorem is as follows.   By hypothesis, there is a sequence $N_j \to \infty$ such that $\frac{|A \cap [-N_j,N_j]|}{|[-N_j,N_j]|}$ converges to a positive limit.   One can define a generalised density of subsets $B \subset {\bf Z}$ by the formula $\mu(B) := \tilde \lim_{j \to \infty} \frac{|B \cap [-N_j,N_j]|}{|[-N_j,N_j]|}$ where $\tilde \lim$ is an extension of the limit functional $\lim$ to bounded sequences (this can be constructed using the Hahn-Banach theorem or using an ultrafilter).   Morally speaking, this turns the integers ${\bf Z}$ into a probability space $({\bf Z},\mu)$ in which $A$ has positive measure and the shift $T: n \mapsto n-1$ is measure-preserving.   Then by the Furstenberg recurrence theorem, for every $k$, there is a positive integer $n$ such that $A \cap T^n A \cap \dots \cap T^{(k-1) n} A$ has positive measure, hence non-empty, hence $A$ contains arbitrarily long arithmetic progressions. 
(I cheated a little because $\mu$ is only a finitely additive measure rather than countably additive, but one can massage the finitely additive probability space $({\bf Z},\mu)$ constructed here into a countably additive model $(X, \tilde \mu)$ by a little bit of measure-theoretic trickery which I will not detail here.) 

A:

Sard's theorem implies that the measure of the set of critical points of a smooth function $f:M_1\to M_2$ between smooth manifolds has measure zero.  Hence the preimage $f^{-1}(x)$ of almost every point in $M_2$ is a smooth submanifold.  This can be used, for example, to prove the existence of Morse functions.  Following Milnor's Morse Theory, Section 6, one can embed $M$ into $\mathbb{R}^n$. Then for almost all points in $\mathbb{R}^n$, the distance map is a Morse function.  This may be seen by applying Sard's theorem to the normal bundle.  The set of focal points has measure zero, and corresponds to the points at which the distance function is degenerate. 

A:

The Chevalley-Warning theorem asserts that if a system of polynomial equations in $r$ variables over a finite field of characteristic $p$ has total degree less than $r$, then the number of solutions to this system is a multiple of $p$.
An immediate corollary of this is Chevalley's theorem: if such a system of polynomials has a "trivial" solution (often this is the origin $(0,\dots,0)$), then it must necessarily have a non-trivial solution as well.   This is often applied for instance as part of the "polynomial method" in combinatorics.