几乎处处收敛和依测度收敛

几乎处处收敛和依测度收敛

几乎处处成立

\[\begin{aligned} \text{ a.e. } &\iff \text{almost everywhere} \iff \text{几乎处处}\\ \text{ a.s. } &\iff \text{almost surely} \iff \text{几乎必然}\\ f \text{ a.e. 有限} &\iff \{f=\pm\infty\} \text{ 是零测集 }\\ f \text{ a.e. 有界} &\iff \exists M>0, \text{ s.t. } \{|f|>M\} \text{ 是零测集 }\\ f \text{ a.e. 可测} &\iff \exists h\in\mathcal{\overline{L}}, \text{ s.t. } f = h \text{ a.e. }\\ f = g \text{ a.e. } &\iff \{f\neq g\} \text{ 是零测集 }\\ f\geq g \text{ a.e. } &\iff \{f < g\} \text{ 是零测集 }\\ \end{aligned} \]

几乎处处收敛到几乎必然收敛

\[设 \{f_n,f,n\geq 1\} \text{ 是 a.e. 有限的广义实值可测函数列} \]

\[\begin{aligned} f_n \to f \text{ a.e. } &\iff \exists \text{ 零测集 } N, \forall \omega\in N^c, \text{ s.t. } f_n(\omega)\to f(\omega)\\ &\iff \forall \varepsilon>0, \mu\left( \bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}\{|f_k-f|\geq \varepsilon\} \right) = 0\\ &\iff \forall \varepsilon>0, \lim_{n\to\infty} \mu\left( \bigcup_{k=n}^{\infty}\{|f_k-f|\geq \varepsilon\} \right) = 0\\ &\iff \forall \varepsilon>0, \lim_{n\to\infty} \mu\left( \sup_{k\geq n}\{|f_k-f|\geq \varepsilon\} \right) = 0\\ \\ \text{R.V. } X_n\to X \text{ a.s. } &\iff \forall \varepsilon>0, P\left(|X_n-X|\geq \varepsilon,\text{ i.o.}\right) = 0 \end{aligned} \]

依测度收敛到依概率收敛

\[\begin{aligned} f_n\xrightarrow{\mu}f &\iff \forall \varepsilon>0, \lim_{n\to\infty} \mu\left( \{|f_k-f|\geq \varepsilon\} \right) = 0\\ &\iff \forall f_{n_k}\subset f_n, \exists f_{n_{k_l}}\subset f_{n_k}, \text{ s.t. } f_{n_{k_l}}\to f\text{ a.e. }\\ \\ \text{R.V. } X_n\xrightarrow{P} X &\iff \forall \varepsilon>0, \lim_{n\to\infty}P\left(||X_n-X||\geq\varepsilon\right) = 0\\ &\iff \forall X_{n_k}\subset X_n, \exists X_{n_{k_l}}\subset X_{n_k}, \text{ s.t. } X_{n_{k_l}}\to X\text{ a.s. }\ \\ \\ X_n\to X\text{ a.s. } &\implies X_n\xrightarrow{P} X \end{aligned} \]