We summarized classical (yet prominent) results in measure theory for easy reference. Please consult the proofs and comprehensive descriptions on the 2nd edition (2008) of Kiyoshi Ito’s book (ISBN978-4-7853-1304-3).
Relationships between various convergences
For a measure space , let be a sequence of -measurable complex functions and a -measurable complex function. Then the followings hold:
|1||converge -a.e.||with s.t. on||If , 1 implies 2.|
If are integrable, 1 implies 4.
|2||converge asymptotically (converge in a Fréchet space ).||,||2 implies 1 for some subsequence .||–|
|3||uniformly converge -a.e. (converge in a Banach space ).||where =||3 implies 1.|
If , 3 implies 4.
|4||converge in mean (converge in )||where||4 implies 2.||are integrable.|
Meanings of classical inequalities
|1||Cauchy–Schwarz||,||Characterize as a Hilbert space (complete norm).|
|2||Hölder’s|| with ,||Product of functions v.s. product of p-norms.|
|3||Minkowski|| with ,||Subadditivity (triangle equality) of norm .|