Approximate limit In some course notes on "evolution equations in probability spaces (etc)" I have the following definition:
A point $z \in \mathbb R^n$ is called the approximate limit of $f$ at $x$ if for every $\epsilon > 0$ the set $\{y \in \Omega : |f(y) - z| \leq \epsilon\}$ has Lebesgue density 1 at $x$. Notation: $\tilde{f}(x) := f(y)$.
I don't understand this definition, what exactly does it say intuitively? The given set also does not depend on $x$... is there something wrong?
 A: I don't know if this will help your intuition, but unwinding the definition a bit, this says that for all $\epsilon\gt0$, for all $\eta\gt0$, there exists $\delta\gt0$ such that in $B_\delta(x)$ you have $|f(y)-z|\leq\epsilon$ outside of a subset of measure less than $\eta\cdot m(B_\delta(x))$.  For a fixed $\epsilon$, letting $\eta$ go to zero makes the relative size of the set where $|f(y)-z|\gt\epsilon$ go to zero (by taking sufficiently small neighborhoods of $x$).  You get an ordinary limit when you don't have to include the exception of a small subset where $|f(y)-z|$ might be larger than $\epsilon$.
In the simplest case, the limit may exist if you ignore the values of $f$ off a set of measure zero.  (E.g., take a continuous function but add $1$ to its values on a countable dense subset of the domain.)  In that case you can actually take $\eta=0$.  Perhaps more illustrative is the example of the function $f$ on $\mathbb{R}$ that is $1$ in an open interval of length $2^{-n}$ centered at $\frac{1}{n}$ for each positive integer $n$, and $0$ elsewhere.  Then the approximate limit of $f$ at $0$ is $0$, but for $\epsilon\lt1$ and $\delta\gt0$, the measure of the subset of $B_\delta(0)$ on which $|f(y)-0|\gt\epsilon$ is positive. 
