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Probability Tidbits 2 - Measure Theory

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Probability Tidbits 2 - Measure Theory

Given a set X, A is a σ-algebra on the set if:

∅,X∈AA∈A⟹Ac∈AAi∈A⟹∪iAi∈A

where the union could be potentially countable. The combination of the set and its σ-algebra (X,A) is called a measurable space. Trivial examples of A include ∅,X and P(X) i.e. the power set. There exists a mapping σ:P(X)→P(X) such that for any subset U⊂X, σ(U) produces the smallest σ-algebra such that U⊂σ(U). So in other words, we can construct a σ-algebra given an arbitrary subset of X.

Also, given a set X, τ is a topology on the set if:

∅,X∈AAi∈τ⟹∪iAi∈τAi∈τ⟹∩NiAi∈τ,N∈N

where the union could be countable but the intersection of sets is finite. The combination of the set and its topology (X,τ) is called a topological space.

Why did I mention these two? In particular, we’re interested in a particular σ-algebra called B(X)=σ(τ) or the Borel σ-algebra on a topology τ of X, which as noted before is defined as the smallest σ-algebra containing τ. We need a measurable space because we’re going to be dealing with probability measures, and we need a topological space because we need some concept of metric in X in order to quantify how close elements in X are with each other (so we can build some analytic intuition on continuity which is important for stochastic calculus).

Given the measurable space (X,A), a map μ:B(X)→[0,∞] is a measure if:

μ(∅)=0∀i≠j,Ai∩Aj=∅⟹μ(∪iAi)=Σiμ(Ai)

Intuitively μ looks really close to P which maps an outcome to a probability, and indeed we’re almost there. Using a change in notation here, probability space (Ω,F,P) is a triple where Ω can be thought of as X, F is a σ-algebra on Ω, and P is our probability measure. In addition to the definition of measure above, we need P(Ω)=1 for it to be a probability measure. When it comes to continuous distributions, we often consider the outcome space Ω to be the continuum, i.e. R, F its corresponding Borel σ-algebra on the typical topology on R. The probability measure is dependent on the probability distribution. For example, the univariate normal distribution measure is defined as P([a,b])=Φ(b)−Φ(a) for any a<b, a,b∈R.

A measurable function f:X→Y is defined as a function that maps one measurable space (X,A) to another measurable space (Y,B) and preserves structure. Specifically, ∀E∈B,f−1(E)∈A. An example of a measurable function is f:R→R,f(x)=x2∀x∈R, since for example f−1((0,9))=(−3,3). These intervals are measurable in the standard (R,B(R)) measurable space. Also, a measurable function is not the same thing as a measure - measurable functions map between two measurable spaces, while a measure maps measurable sets of a measurable space into the positive extended real number line!

In this sense random variables are measurable functions mapping from Ω, the set in a probability space (Ω,F,P), to R (typically). The standard notation of random variable equaling a real value x can thus be thought of as the preimage of the random variable on x:

P(X=x,x∈R)=P({ω:X(ω)=x})=P(X−1(x))

Supposed the preimage was n elements of the set Ω:

X−1(x)={ω1,ω2,...,ωn}

Then, someone could also just say:

P({ω1,ω2,...,ωn})=P(X−1(x))


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