Measure theory: convergences and inequalities summary

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 (X,\mathcal{B},\mu), let \{f_n\} be a sequence of \mu-measurable complex functions and f a \mu-measurable complex function. Then the followings hold:

NoNotationDefinitionImplicationsExtra Conditions
1converge \mu-a.e.\exists E_0\subset X with \mu(E_0)=0 s.t. \lim_n f_n = f on X-E_0.If \mu(X)<\infty, 1 implies 2.
If f,\,\{f_n\} are integrable, 1 implies 4.
2converge asymptotically (converge in a Fréchet space S(X)).\forall \epsilon>0,
\displaystyle\lim_{n\to \infty}\mu(X(|f_n-f|>\epsilon))=0.
2 implies 1 for some subsequence \{f_{n(k)}\}.
3uniformly converge \mu-a.e. (converge in a Banach space M(X)).\|f_n-f\|_\infty\to 0\,(n\to\infty) where \|f\|_\infty={\rm ess.sup}_{x\in X}|f|=\inf \{\alpha\geq 0:|f|\leq \alpha\ a.e.\}.3 implies 1.
If \mu(X)<\infty, 3 implies 4.
4converge in mean (converge in L^p)\|f_n-f\|_p\to 0\ (n\to \infty) where \displaystyle\|f\|_p=\{\int |f(x)|^p\,d\mu\big\}^{1/p}.4 implies 2. f,\,\{f_n\} are integrable.

Meanings of classical inequalities

1Cauchy–Schwarz\forall f,\,g\in L^2,
|(f,g)|^2\leq \|f\|^2_2\cdot \|g\|^2_2.
Characterize L^2 as a Hilbert space (complete norm).
2Hölder’s\forall f\in L^p,\,g\in L^q with p>0,\,\frac{1}{p}+\frac{1}{q}=1,
\displaystyle|\int fg\,dx|\leq \|f\|_p\cdot \|g\|_q.
Product of functions v.s. product of p-norms.
3Minkowski\forall f,\,g\in L^p with p\geq 1,
\|f+g\|_p\leq \|g\|_p+\|g\|_p.
Subadditivity (triangle equality) of norm \|\cdot \|_p.