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A probability space is a measure space $(\Omega, \mathcal{F}, P)$ where $\Omega$ is the sample space, $\mathcal{F}$ is a sigma-algebra of events, and $P$ is a probability measure with $P(\Omega) = 1$.

A sigma-algebra $\mathcal{F}$ over a set $\Omega$ is a collection of subsets of $\Omega$ that includes the empty set, is closed under complementation, and is closed under countable unions.

A probability measure $P$ is a function from a sigma-algebra $\mathcal{F}$ to the interval $[0,1]$ such that $P(\Omega) = 1$, it is countably additive, and $P(\emptyset) = 0$.

A random variable is a measurable function from a probability space into a measurable space, typically $(\Omega, \mathcal{F}) \mapsto (\mathbb{R}, \mathcal{B}(\mathbb{R}))$, where $\mathcal{B}(\mathbb{R})$ is the Borel sigma-algebra on the real numbers.

The expected value of a random variable $X$, denoted $E[X]$, is the integral $\int_{\Omega} X dP$ in a probability space $(\Omega, \mathcal{F}, P)$, providing a measure of the 'center' of the distribution of $X$.

The variance of a random variable $X$ is a measure of the spread of its distribution, defined as $Var(X) = E[(X - E[X])^2]$.

Two events $A$ and $B$ in a probability space are independent if $P(A \cap B) = P(A)P(B)$. For random variables, independence means the joint distribution factors into the product of the marginals.

Given two events $A$ and $B$ with $P(B) > 0$, the conditional probability $P(A|B)$ is defined as $\frac{P(A \cap B)}{P(B)}$.

If $\{B_n\}_{n=1}^{\infty}$ is a countable partition of the sample space $\Omega$, then for any event $A$ in $\Omega$, $P(A) = \sum_{n} P(A|B_n)P(B_n)$ given that each $P(B_n) > 0$.

If $\{A_n\}$ is a sequence of events in a probability space, then if the sum of the probabilities is finite, i.e., $\sum_{n=1}^{\infty} P(A_n) < \infty$, the probability that infinitely many of them occur is 0.

A sequence of random variables $X_n$ converges almost surely to a random variable $X$ if $P(\lim_{n \to \infty} X_n = X) = 1$. It implies that the set of $\omega$ for which $X_n(\omega)$ does not converge to $X(\omega)$ has probability zero.

A sequence of random variables $X_n$ converges in probability to a random variable $X$ if for every $\epsilon > 0$, $\lim_{n \to \infty} P(|X_n - X| > \epsilon) = 0$.

A sequence of random variables $X_n$ converges in distribution to a random variable $X$ if for every continuity point $x$ of the distribution function $F_X$, $\lim_{n \to \infty} F_{X_n}(x) = F_X(x)$, where $F_{X_n}$ and $F_X$ are the respective cumulative distribution functions.

The $L^p$-space is a vector space of equivalence classes of measurable functions for which the $p$-th power of the absolute value is integrable, defined by the norm $||f||_p = ( \int |f|^p dP )^{1/p}$.

Fatou's Lemma states that for any sequence of non-negative measurable functions $\{f_n\}$, $\int \liminf_{n \to \infty} f_n dP \leq \liminf_{n \to \infty} \int f_n dP$.

The Monotone Convergence Theorem states that if $\{f_n\}$ is a sequence of non-negative measurable functions increasing to $f$, then $\lim_{n \to \infty} \int f_n dP = \int f dP$.

The Dominated Convergence Theorem states that if $\{f_n\}$ are measurable, $f_n \to f$ a.e., and there exists an integrable function $g$ such that $|f_n| \leq g$ a.e., then $\lim_{n \to \infty} \int f_n dP = \int f dP$.

The Radon-Nikodym Theorem states that if $P$ and $Q$ are two sigma-finite measures on a space $(\Omega, \mathcal{F})$, and $Q$ is absolutely continuous with respect to $P$, there exists a measurable function $f$ such that $Q(A) = \int_A f dP$ for all $A\in \mathcal{F}$.