 The lemma plays an important role in the conditional expectation in probability theory, where it allows replacement of the conditioning on a random variable by conditioning on the formula_1-algebra that is generated by the random variable.In the lemma below, formula_4 is the extended real number line, and formula_5 is the formula_1-algebra of Borel sets on formula_7
The notation formula_8 indicates that formula_9 is a function from formula_10 to formula_11 and that formula_9 is measurable relative to the formula_1-algebras formula_14 and formula_15

Furthermore, if formula_16 and formula_17 is a measurable space, we define
One can easily check that formula_19 is the minimal formula_1-algebra on formula_10 under which formula_22 is measurable, i.e.Let formula_24 be a function from a set formula_25 to a measurable space formula_26 and formula_27 is formula_28-measurable.