ABSTRACT
We start with an example. Let μ represent Lebesgue measure on the interval [−1, 1] and let https://www.w3.org/1998/Math/MathML"> B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq2319.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> denote the corresponding set of Lebesgue measurable subsets of [−1, 1]. Let https://www.w3.org/1998/Math/MathML"> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq2320.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> denote the subset of https://www.w3.org/1998/Math/MathML"> B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq2321.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> consisting of those Lebesgue measurable subsets of [−1, 1] which are symmetric with the origin; i.e., Y ∈ https://www.w3.org/1998/Math/MathML"> B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq2322.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is in https://www.w3.org/1998/Math/MathML"> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq2323.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if for each x ∈ [−1, 1], x ∈ Y if and only if −x ∈ Y. Now any real valued function whose domain is symmetric with respect to the origin can be written uniquely as a sum of an even function and an odd function, one simply uses the functions f e (x) = (f(x) + f(−x))/2 and f o (x) = (f(x) − f(−x))/2. If f is integrable on [−1, 1], then we have that ∫ Y f dμ = ∫ Y f e dμ since the integral of an odd (integrable) function is zero on any measurable set which is symmetric with the origin. In such a case, f e is said to be the “conditional expectation of f with respect to https://www.w3.org/1998/Math/MathML"> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq2324.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ”. We will revisit this example in the second section.
