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Random Variables

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Measurable Functions

Let (S, &Sigma. ), (Q, &Phi. ) be measurable spaces. A function ~h#: S &rightarrow. Q is ~{measurable} (w.r.t. &Sigma. , &Phi. ) _ if ~h^{-1}(A) &in. &Sigma., _ &forall. A &in. &Phi..
[Write ~h^{-1}(&Phi.) &subseteq. &Sigma. ].

The map ~h^{-1} preserves set operations:

  • ~h^{-1}(&union._{~j} A_{~j}) = &union._{~j} ~h^{-1}(A_{~j})
  • ~h^{-1}(&intersect._{~j} A_{~j}) = &intersect._{~j} ~h^{-1}(A_{~j})
  • ~h^{-1}(A^{~c}) = (~h^{-1}(A))^{~c}
  • etc.

So if ~h is measurable, the range of ~h^{-1}, that is ~h^{-1}(&Phi.), is itself a sub-&sigma.-algebra of &Sigma..

Real-valued Measurable Functions

We will now consider functions such as ~h#: S &rightarrow. &reals., _ which are measurable (w.r.t. &Sigma., @B).
Write m&Sigma. for the class of &Sigma.-measurable functions on S, and (m&Sigma.)^+ for the non-negative elements of m&Sigma..

Borel Functions

~h#: S &rightarrow. &reals., _ S topological space, is called ~{Borel} if ~h is @B(S)-measurable. Most important case is ~h#: &reals. &rightarrow. &reals.

Propositions on Measurability

  • If _ @C &subseteq. @B, _ and &sigma.(@C) = @B, then _ ~h^{&minus.1}#: @C &rightarrow. &Sigma. _ &imply. _ ~h &in. m&Sigma.
  • S topological space and _ ~h#: S &rightarrow. &reals. _ continuous _ then ~h is Borel
  • For any measurable space (S, &Sigma.) a function _ ~h#: S &rightarrow. &reals. _ is &Sigma.-measurable if

    \{ ~h &le. ~c \} #:= \{ ~s &in. S | ~h(~s) &le. ~c \} &in. &Sigma. _ _ (&forall. ~c &in. &reals.)

Sums and Products of Measurable Functions

m&Sigma. is and algebra over &reals., i.e. if λ &in. &reals. and ~h_1, ~h_2, ~h_3 &in. m&Sigma., then:

  • ~h_1 + ~h_2 &in. m&Sigma., _ ~h_1~h_2 &in. m&Sigma., _ λ _ ~h_1 &in. m&Sigma.

Composition Lemma

 

Measurability of Limits

 

Random Variables

(&Omega., @F, P) probability space, (Q, &Phi. ) measurable space. A ~{random variable} is a measurable function _ X#: (&Omega., @F ) &rightarrow. (Q, &Phi. ).
(Q,&Phi. ) is called the ~{state space} or ~{range space} of the random variable.

Real-valued Random Variables

From now on we will be dealing primarily with the special case (Q, &Phi. ) = ( &reals., @B )

Distribution of Random Variable

X#: (&Omega., @F ) &rightarrow. (Q,&Phi. ) random variable. The ~{distribution}, or ~{law} of X (with respect to P) is the function P^X #: &Phi. &rightarrow. [0,1],

P^X (A) _ = _ P (\{&omega. | X(&omega.) &in. A\}) _ = _ P (X^{-1}(A)) This is sometimes written in the form P(X &in. A).