Relation between the types of convergences of sequences in $\mathbb{L}_p$ Spaces.

Question

As far as I know there are 4 types of convergence in $\mathbb{L}_p$ spaces.

1. Pointwise Convergent a.e

2. Uniformly Convergent almost a.e

3. Convergence in Measure

4. p-Convergence

My question is how they are related? Are there any more types of convergence in Function Spaces?

Answer 1

I looked online for a single diagram picturing all the modes of convergence and couldn't find one. The only available one was mentioned in the comments and comes as separate diagrams. Back in my college days, my measure theory professor, Guillermo Grabinsky, handed us a handwritten version of the following diagram. All the credit goes to him, I simply used TiKz to type it and post it here.

Definitions

• $(X, \mu)$ is a measure space.
• $p \in [1, \infty)$.
• $L^p:=L^p(X, \mu)$ and $L^\infty:=L^\infty(X,\mu)$.
• $a.e.$ := almost everywhere.
• $u.$ := uniform.
• $a.u$ := almost uniform.
• $\mu$:= in measure.
• $C.$ := Cauchy.

Arrows

1. $\longrightarrow$ Requires $\mu(X)< \infty$.
2. $\dashrightarrow$ Existence of a subsequence that converges in the mode at which the arrow points.
3. $\Longrightarrow$ Implication with no restrictions.