1.2.1 Definitions

The family of r.v. X: ? ? X ( ?)

( ?, a, P) ( )

forms a vector space on , denoted ?.

Two vector subspaces of ? play a particularly important role and these are what will be defined.

The definitions would be in effect the final element in the construction of the Lebesgue integral of measurable mappings, but this construction will not be given here and we will be able to progress without it.

DEFINITION. We say that two random variables X and X ? defined on ( ?, a) are almost surely equal and we write X = X ? a.s. if X = X ? except eventually on an event N of zero probability (that is to say N ? a and P ( N) = 0).

We note:

• = {class (of equivalences) of r.v. X ? almost definitely equal to X};

• = {class (of equivalences) of r.v. almost definitely equal to O}.

We can now give:

• the definition of L ¹ ( dP) as a vector space of first order random variables; and

• the definition of L ² ( dP) as a vector space of second order random variables:

  
  

where, in these expressions, the r.v. are clearly...