## Borel Measurability

Let g be a real function and let X be a random variable defined on the probability space {^, P}. For g(X) to be a random variable, we must have that

It is possible to construct a real function g and a random variable X for which this is not the case. But if

For all Borel sets B, Ab = {x є R : g(x) є B} is a Borel set itself,

(2.5)

then (2.4) is clearly satisfied because then, for any Borel set B and A B defined in (2.5),

{ш є ^ : g(X(o)) є B} = {o є ^ : X(o) є AB}є

Moreover, if (2.5) is not satisfied in the sense that there exists a Borel set B for which A B is not a Borel set itself, then it is possible to construct a random variable X such that the set

{o є ^ : g(X(ш)) є B} = {o є ^ : X(ш) є AB} /

hence, for such a random variable X, g(X) is not a random variable itself...

Read More