Denote our measure space as . Note that a property p holds for almost every if the set has measure zero.
- A sequence of measurable functions converges pointwise if for all .
- A sequence of measureable functions converges uniformly if .
- A sequence of measurable functions converges to pointwise almost everywhere if for almost every , or .
- A sequence of measurable functions converges in if
check Convergence in Measure for convergence in measure.
Relevant Properties 
- through uniform convergence implies through pointwise convergence, which in turn implies pointwise a.e. convergence.
- through convergence implies through pointwise a.e convergence up to a subsequence.
- Pointwise a.e. convergence, equipped with dominating function, implies in . To see the proof and examples of why we need a dominating function, read Dominated Convergence Theorem.
- Convergence in Measure lists relationships between convergence in measure and other forms of convergence.
Proof implies in measure
We can see that that
we can see by convergence that
Hence the proposition follows.
Proof through convergence implies pointwise a.e convergence up to a subsequence.
convergence means in measure. Because in measure, there exists a subsequence such that pointwise a.e.
Example: in measure does not imply convergence in
Let . Then in measure, but does not converge to 0 in . This is because for all .
- ↑ Craig, Katy. MATH 201A Lecture 17. UC Santa Barbara, Fall 2020.
- ↑ Craig, Katy. MATH 201A Lecture 18. UC Santa Barbara, Fall 2020.