# Corresponding convergence types of pointwise and uniform convergence of calculus in measure theory

What are the related convergence types of pointwise and uniform convergence definitions of calculus in measure theory?

As we know from calculus, uniform convergence is $stronger$ than pointwise convergence. Also we can say that convergence in measure is the $weakest$ type of all convergences(e.g almost everywhere convergence, almost uniformly convergence etc.)

I think uniform convergence in calculus corresponds to uniform convergence in measure theory. To add, pointwise convergence in calculus corresponds to almost uniform convergence in measure theory.

First of all, one of your statements is false: there are weaker kinds of convergence even than convergence in measure.

Anyway, there is not a linear hierarchy here, although it's close in a finite measure space. I have roughly organized this list to be in order of strength.

There is weak $L^p$ convergence. This is a general Banach space notion applied to the special case of $L^p$. Specifically, if $p \in [1,\infty)$, then $f_n \to f$ weakly in $L^p(X,\mu)$ if and only if for any $g \in L^q(X,\mu)$, $\int_X f_n g d \mu \to \int_X f g d \mu$ (in the sense of real numbers). Here $q=\frac{p}{p-1}$ for $p>1$, $q=\infty$ for $p=1$. I have used the Riesz representation theorem here to identify the bounded linear functionals on $L^p$ as integration against functions in $L^q$. Parseval's identity implies that that $f_n(x) = \sin(nx)$ on $[0,\pi]$ converges weakly in $L^2$ to $0$; this shows that weak convergence in $L^p$ can be weaker than convergence in measure.

There is convergence in measure.

There is strong $L^p$ convergence.

There is pointwise convergence and a.e. convergence. Pointwise convergence is of little use in measure theory; a.e. convergence is the more natural notion. This is, surprisingly, the oddball among convergence types: conditions relating a.e. convergence to the other convergence types are the most complicated.

There is almost uniform convergence.

There is uniform convergence off a null set. This is usually called (strong) $L^\infty$ convergence. This is, perhaps surprisingly, strictly stronger than almost uniform convergence, as the example $f_n(x)=x^n$ on $[0,1]$ shows.

There is uniform convergence. Like pointwise convergence, this is not a convenient notion for measure theory.

These don't really have nice correspondences to the notions from advanced calculus. What sorts of correspondences were you hoping to draw? Or are you interested in a summary of the conditions relating the different convergence types?

I'm not sure if this is what you're asking, but the notions are the same; i.e. "pointwise convergence" means that for all $\epsilon> 0$, all $x$, there is $N$ so that $|f_n(x) - f(x)| < \epsilon$ if $n > N$, whether you're studying measure theory or not.

There is the idea of "pointwise a.e. convergence", meaning for all $\epsilon> 0$, a.e. $x$, there is $N$ so that $|f_n(x) - f(x)| < \epsilon$ if $n > N$, and of course, "a.e." is a measure-theoretic definition (it depends on the measure in question). This is useful for statements along the lines of

"If $f_n \to f$ in measure, then there is a subsequence $f_{n_k}$ that converges pointwiwse a.e. to $f$."

This property is also sometimes called "almost everywhere convergence".

There is another property, "almost uniform convergence": for every $\delta > 0$, there exists a set $X_\delta$ with measure less than $\delta$ such that $f_n$ converges uniformly on $X \backslash X_\delta$. (This is different than "a.e. uniform convergence" which would mean that for a fixed, zero-measure set $X_0$, $f_n$ converges uniformly on $X \backslash X_0$).

Wikipedia has a list of types of convergence (and their relative strength): http://en.wikipedia.org/wiki/Modes_of_convergence_%28annotated_index%29