A Space

So, an article not by AI. Can people differ which is by the AI.
AI -> 100%. Human -> 61%. Afford -> 100% in you. and the work is not for been read.
only typing, kokoro is important.

A Space

• Together with a triple (or collection) of axioms which define operations on and relations between those objects.
• A triple (or collection) of objects

Measurable Space

\((\Omega, \mathcal{F})\)

• $\Omega$: set.

• \(\mathcal{F} \subseteq \mathcal{P}(\Omega)\) is a (\sigma)-algebra:

  1. \(\Omega \in \mathcal{F}\)
  2. \(A \in \mathcal{F} \implies A^c \in \mathcal{F}\)
  3. \(A_1, A_2, \ldots \in \mathcal{F} \implies \bigcup_{i=1}^{\infty} A_i \in \mathcal{F}\)

A specific measurable space: Borel Space

\((\mathbb{R}, \mathcal{B}(\mathbb{R}))\)

• \(\mathbb{R}\): set of real numbers.
• \(\mathcal{B}(\mathbb{R}) = \sigma(\{ (a, b) : a, b \in \mathbb{R}, a < b \})\) is the Borel \(\sigma\)-algebra (the smallest \(\sigma\)-algebra containing all open intervals).

Measure Space

\((X, \Sigma, \mu)\)

• \((X, \Sigma)\) is a measurable space.

• \(\mu: \Sigma \to [0, +\infty]\) is a measure satisfying:

  1. \(\mu(\emptyset) = 0\)
  2. For any disjoint sequence \(\{A_n\}_{n=1}^{\infty} \subset \Sigma\): $ \mu\left( \bigcup_{n=1}^{\infty} A_n \right) = \sum_{n=1}^{\infty} \mu(A_n)$

Probability Space

\((\Omega, \mathcal{F}, Pr)\)

• \((\Omega, \mathcal{F})\) is a measurable space.

• \(Pr: \Sigma \to [0, 1]\) is a probability measure satisfying:

  1. \(Pr(\emptyset) = 0\)
  2. For any disjoint sequence \(\{A_n\}_{n=1}^{\infty} \subset \Sigma\): $ \mu\left( \bigcup_{n=1}^{\infty} A_n \right) = \sum_{n=1}^{\infty} \mu(A_n)$

\(Pr(\Omega)=1\)