Product measure theorems, Define T , the "ordinal

Product measure theorems

Reference no: EM13966768

1. Let (X, B, µ) be σ-finite and f any measurable real-valued function on X . Prove that (µ × λ){(x, y): y = f (x )}= 0 (the graph of a real measurable function has measure 0).

2. Let (X, ≤) be an uncountable well-ordered set such that for any y ∈ X, {x ∈ X : x y} is countable. For any A ⊂ X , let if µ( A) = 0 if A is countable and µ( A) = 1 if X \ A is countable. Show that µ is a measure defined on a σ -algebra. Define T , the "ordinal triangle," by T := {(x , y) ∈ X × X : y <>x }. Evaluate the iterated integrals (( 1T (x, y) dµ(y) dµ(x ) and (( 1T (x, y) dµ(x ) dµ(y). How are the results consistent with the product measure theorems?

Reference no: EM13966768

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