Tao, Terence, Topics in Random Matrix Theory

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MTIME: [2025-04-08 Tue 12:41],[2025-04-08 Tue 12:37]

Suppose a \(\sigma\)-algebra \(\Sigma\) over \(X\) . A measure is a function \(\mu: \Sigma \to \bb{R}\), with the following properties:

• non-negativity: for all \(E \in \Sigma, \mu(E) \ge 0\)
• \(\mu(\emptyset) = 0\)
• countable additivity: for all countable collections \(\{E_k\}_{k=1}^{\infty}\) of pairwise disjoint sets in in \(\Sigma\), \(\mu(\bigcup_{k=1}^{\infty}E_k) = \sum_{k=0}^{\infty}\mu(E_k)\)

\((X, \Sigma)\) is called a measurable space. Members of Σ are measurable sets. \((X, \Sigma, \mu)\) is a measure space.

A probability measure is a measure with \(\mu(\Sigma) = 1\).

This is the wikipedia definition. Tao uses \((\Omega, \mathcal{B}, \mathbb{P})\) instead of \( (X, \Sigma, \mu)\):

• sample space \(\Omega\)
• \(\mathcal{B}\) , a \(\sigma\)-algebra over \(\Omega\)
  • We'll call subsets of \(\Omega\), i.e. members of \(\mathcal{B}\), events
• Probability measure \(\mathbb(P)\) on the sample space:
  • \(E \mapsto \mathbb{P}(E)\) where \(E \subset \Omega\) i.e. E is an event
  • \(\mathbb{P}(E) \in [0,1]\) - non-negativity
  • \(\mathbb{P}\) is countably additive.
  • \(\mathbb{P}(Omega) = 1\) - together with coutable additivity, implies \mathbb{P}(∅) = 0.

We'll notate elements of \(\Omega\) as \(\omega\), but want to avoid referring to these directly as much as possible. Why?

REVIEW_SCORE: 3.0
MTIME: [2025-04-08 Tue 12:41],[2025-04-08 Tue 12:41]

We avoid referring to elememnts of Ω directly because we're trying to study the type family of probability measures, and especially extensions of the measure space.

\((\Omega^0, \mathcal{B}^0, \mathbb{P}^0)\) extends \((\Omega, \mathcal{B}, \mathbb{P})\) if there exists surjective map \(\pi: \Omega^0 \to \Omega\) that is

• measurable, i.e.\(\pi^{-1} (E) \in \mathcal{B}^0\) for every \(E \in \mathcal{B}\)
• probability preserving, i.e. \(\mathbb{P}^0(\pi^{-1}(E)) = \mathbb{P}(E)\) for every \(E \ in mathcal{B}\)

\(\pi\) ends up identifying every event \(E \in \mathcal{B}\) with a \(\pi^{-1}(E \in \mathcal{B}^0\).

alternate source for this calls π a probability preserving measurable map.

(there's what appears to be a minor abuse of terminology that seems common here, where \(\pi : \Omega^0 \to \Omega\) but \(\pi^{-1}\) is applied to events E, not samples \(\omega\). This is just asking for a preimage made by a surjective function, so I guess that's okay. That's how the blog entry seems to use \(\pi\), anyway.)

We mostly therefore want to frame our explorations and conclusions in terms of the bits that will let us generalize \(\Omega\).

If we fail to do this, let's consider it as no longer doing probability theory. This subject is demarcated by the type family; once you're looking at properties of a specific member that don't generalize across extensions, you're looking at a different thing. In other words, this is our invariance.

\begin{equation} \mathbb{P}(\bigvee_{i}~E_i) \leq \sum_{i}~\mathbb{P}(E_i) \end{equation}

for any countable collection of events E_i .

take complements:

\begin{equation} \mathbb{P}(\overline{\bigwedge_{i}~E_i}) \leq \mathbb{P}(\overline{E_i}) \end{equation}