The Earlier and the New Versions of the Definitions of Probability Spaces, Laws, Expectation, Independence and Modes of Convergence Agree

In the setting of Probability Spaces: Notation.

(i) Probability spaces and random elements. For every measurable space $(Z,\mathcal{A})$ and every measure $\mu$ on it, $(Z,\mathcal{A},\mu)$ is a probability space in the earlier version if and only if it is a probability space in the new version. The two versions give the same events of $(\Omega,\mathcal{F},P)$ (Probability Spaces, Events and Random Elements (ii)); and for every metric space $(S,d)$ and every $n\in\mathbb{N}$, a map $X:\Omega\to S$ is a random element of $(S,d)$ in the earlier version if and only if it is a random element of $(S,d)$ in the new version, a map $X:\Omega\to\mathbb{R}$ is a random variable in the earlier version if and only if it is a random variable in the new version, and a map $X:\Omega\to\mathbb{R}^{n}$ is a random vector in $\mathbb{R}^{n}$ in the earlier version if and only if it is a random vector in $\mathbb{R}^{n}$ in the new version.

(ii) Laws. For every metric space $(S,d)$ and every random element $X$ of $S$, the law of $X$ in the earlier version is its law in the new version; and for every probability space $(\Omega',\mathcal{F}',P')$ and every random element $Y$ of $S$ on it (in either version, by (i)), $X$ and $Y$ are identically distributed in the earlier version if and only if they are identically distributed in the new version.

(iii) Expectation. For all $n,p,q\in\mathbb{N}$, the earlier and the new version give the same expectation $\mathbb{E}X$ of every $X\in\mathcal{M}^{+}(\Omega,\mathcal{F})$ (Expectation (i), Expectation (i)) and of every $X\in\mathcal{L}^{1}(P;\mathbb{R}^{p\times q})$ (Expectation (iv), Expectation (iv)); and a random variable, or a random vector in $\mathbb{R}^{n}$, is integrable in the earlier version if and only if it is in the new version, with the same expectation (Expectation (ii), Expectation (ii); Expectation (iii), Expectation (iii)).

(iv) Independence. For every set $J$, a family $(A_{j})_{j\in J}$ of events is independent in the earlier version if and only if it is in the new version (Independence of Events and of Random Elements (i)); and for every family $((S_{j},d_{j}))_{j\in J}$ of metric spaces and every family $(X_{j})_{j\in J}$ in which each $X_{j}$ is a random element of $S_{j}$, the family $(X_{j})_{j\in J}$ is independent in the earlier version if and only if it is in the new version (Independence of Events and of Random Elements (ii)). For all metric spaces $(S,d)$ and $(T,e)$, every random element $X$ of $S$ and every random element $Y$ of $T$, $X$ and $Y$ are independent in the earlier version if and only if they are in the new version (Independence of Events and of Random Elements (iii)). For every metric space $(S,d)$, every random element $X$ of $S$ and every $\sigma$-algebra $\mathcal{G}$ on $\Omega$ contained in $\mathcal{F}$, $X$ is independent of $\mathcal{G}$ in the earlier version if and only if it is in the new version (Independence of Events and of Random Elements (iv)). For every metric space $(S,d)$ and every sequence $(X_{k})_{k\in\mathbb{N}}$ of random elements of $S$, $(X_{k})$ is independent and identically distributed in the earlier version if and only if it is in the new version (Independence of Events and of Random Elements (v)).

(v) Convergence in probability. For every metric space $(S,d)$ that is separable, in either version by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (ii), every sequence $(X_{n})$ of random elements of $S$ and every random element $X$ of $S$, $(X_{n})$ converges to $X$ in probability in the earlier version if and only if it converges to $X$ in probability in the new version.

(vi) Almost sure convergence. For every metric space $(S,d)$, every sequence $(X_{n})$ of random elements of $S$ and every random element $X$ of $S$, $(X_{n})$ converges to $X$ almost surely in the earlier version if and only if it converges to $X$ almost surely in the new version.

(vii) Convergence in $L^{p}$. For every $k\in\mathbb{N}$, every real $p\ge1$, every sequence $(Y_{n})$ of random vectors in $\mathbb{R}^{k}$ and every random vector $Y$ in $\mathbb{R}^{k}$ with $\lVert Y_{n}-Y\rVert\in\mathcal{L}^{p}(P)$ for every $n\in\mathbb{N}$, where $\lVert\cdot\rVert$ is the Euclidean norm, $(Y_{n})$ converges to $Y$ in $L^{p}$ in the earlier version if and only if it converges to $Y$ in $L^{p}$ in the new version.

(viii) Weak convergence. For every metric space $(S,d)$, every sequence $(\mu_{n})$ of finite measures on $(S,\mathcal{B}(S))$ and every finite measure $\mu$ on $(S,\mathcal{B}(S))$, $(\mu_{n})$ converges weakly to $\mu$ in the earlier version if and only if it converges weakly to $\mu$ in the new version.

(ix) Convergence in distribution. For every metric space $(S,d)$, every sequence $((\Omega_{n},\mathcal{F}_{n},P_{n}))_{n\in\mathbb{N}}$ of probability spaces, every sequence $(X_{n})$ in which each $X_{n}$ is a random element of $S$ on $(\Omega_{n},\mathcal{F}_{n},P_{n})$, and every random element $X$ of $S$ (in either version, by (i)), $(X_{n})$ converges to $X$ in distribution in the earlier version if and only if it converges to $X$ in distribution in the new version.