Random Variable
Measurable function Let $(E, \mathcal{E})$ and $(F, \mathcal{F})$ be measurable spaces where $E,F,$ are sets and $\mathcal{E}$ and $\mathcal{F}$ are $\sigma...
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Probability Space
The sample space $\Omega$ is the set of all possible outcomes of an experiment. Points in $\omega \in \Omega$ are outcomes or realizations.
Limit Superior and Inferior of a Sequence
Definition 2.5.1 Let $\{s_n\}$ be a sequence in $\mathbb{R}$. The limit superior of $\{s_n\}$, denoted as $\limsup_{n\to\infty}s_n$, is defined as
Subsequences and Bolzano-Weierstrass Theorem
Definition 2.4.1 Let \(\{p_n\}\) be a sequence and let \(\{n_k\}\) be strictly increasing sequence, i.e. $n_1 < n_2 <n_3 < \cdots$. We call \(\{p_{n...
Monotone Sequences
Definition 2.3.1 A sequence \(\{a_n\}\) is said to be (a) monotone increasing if $a_n \leq a_{n+1}$ for all $n\in\mathbb{N}$ (b) monotone decreasing if $a_...