Tao, Terence, Topics in Random Matrix Theory
- authors
- Tao, Terence
Reading the introductory section of this on Ed's recommendation for a quick overview of probability theory, i.e. just section 1.1. Running ntoes for now, expansion as warranted.
opening remarks
assuming familiarity with measure theoy, eh? okay, let's see how badly this screws us over.
Foundations
what is a measure?
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.
notation
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?
Probability theory studies properties of probability measures that are invariant across extensions of the probability space.
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.
what is preserved under extension?
preserved:
\mathbb{P}(E)
E ?= \emptyset
E ?= F
notice that this needs surjectivity. why?
otherwise there might be an\omega \in \Omega s.t. E \cupdot F = \{\omega\}, but \pi^{-1}\{\omega\} = \emptyset and so \pi^{-1} E = \pi^{-1} F even though E \neq F.
could you drop surjectivity and still have the map be measurable and probability-preserving?
\cup, \cap, \complement
not preserved:
cardinality of an event - e.g. \Omega = {1...6} , \Omega^0 = {1...6} \times {1...6} when you "add a die"
notation to reflect leaving the underlying set behind: logical language, not set theoretic.
we'll use \lor, \land,\overline instead of \cup, cap,\complement
we'll say events "hold". E \lor F is "the event that E or F hold".
still using \subset , though. We're calling it "contains", "implies", or "only if".
eqn 1.1 the union bound.
for any countable collection of events Ei .
take complements: