Calculus in Gauss Space

The �-dimensional Lebesgue space is the measurable space (E���(E�))— where E = [0 � 1) or E = R—endowed with the Lebesgue measure, and the “calculus of functions” on Lebesgue space is just “real and harmonic analysis.” The �-dimensional Gauss space is the same measure space (R���(R�)) as in the previous paragraph, but now we endow that space with the Gauss measure P� in place of the Lebesgue measure. Since the Gauss space (R���(R�) �P�) is a probability space, we can—and frequently will— think of any measurable function � : R� → R as a random variable. Therefore, P{� ∈ A} = P�{� ∈ A} = P�{� ∈ R� : � (�) ∈ A}�