My friends and I are studying for an exam in measure theory and are looking for some interesting/odd functions in measure theory that might be useful as examples or counterexamples.
Some properties we are dealing with include:
$f_n\to f$ pointwise a.e.: $\mu(\{x\in X\mid \limsup|f_n(x)-f(x)|>0\})=0$
$f_n\to f$ in measure: $\forall\varepsilon>0,$ $\lim\limits_{n\to\infty}\mu(\{x\in X\mid |f_n(x)-f(x)|>\varepsilon\})=0$
$f_n\to f$ almost uniformly: $\forall \varepsilon>0,$ $\exists X_\varepsilon$ such that $\mu(X\setminus X_\varepsilon)<\varepsilon$ and $\lim\limits_{n\to\infty}\sup\limits_{x\in X_\varepsilon}|f_n(x)-f(x)|=0$
$f_n\to f$ uniformly a.e.: $\exists X'$ such that $\mu(X\setminus X')=0$ and $\lim\limits_{n\to\infty}\sup\limits_{x\in X'}|f_n(x)-f(x)|=0$
Specifically, the first two and latter two types of convergence seem the same, and we are wondering how they differ. Besides that, any interesting function is worth at least an upvote.
Convergence in measure vs pointwise a.e. convergence:
One can have convergence in measure with pointwise convergence nowhere. The main idea of the example is to have a function whose support shrinks and simultaneously sweeps across the domain repeatedly. Explicitly writing out this example is not so instructive but it is done in most textbooks.
This issue is definitely counterintuitive. It can be understood a bit better by using the Borel-Cantelli lemma to prove that if the convergence in measure is sufficiently rapid, then almost everywhere convergence is ensured.
Uniform convergence vs almost uniform convergence:
Take a sequence of continuous functions on a compact interval converging pointwise to a function with a jump discontinuity and no other discontinuities. Cut out an arbitrarily small interval around the discontinuity and you will have uniform convergence. But you do not have uniform convergence originally. This is again done explicitly in most textbooks.
This issue might be counterintuitive at first, since it seems like you should be able to just "send the measure of the small set to zero" and obtain uniform convergence. But notice that Egorov's theorem is the main place where the notion of almost uniform convergence comes into play; other than that, it's not really relevant. The lack of conditions on Egorov's theorem shows you why almost uniform convergence must be strictly weaker than uniform convergence: if it were not, then any pointwise and uniform limits on compact sets would be exactly the same, which we know is not the case.
An issue similar to this one with Egorov's theorem pops up with Lusin's theorem, in which excising these small measure sets has topological implications (it "disconnects" non-commensurable pieces of a discontinuous function).