Errata: "Pullbacks and Coherent-Like Properties" [In: Lect. Notes Pure Appl. Math., Dekker, 205 (1999) 437-459]
Thomas G. Lucas · 2017
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.