Talk:Product measure

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Is the uniqueness of product measure guarantee only if in cases that both (X₁,Σ₁,μ₁) and (X₂,Σ₂,μ₂) are σ-finite? -- Jung dalglish 16:20, 8 February 2006 (UTC)

You aer right. A counterexample to the fubini theorem where one of the measures is not sigma finite is, for example, borel measure on [0, 1] times the counting measure on [0,1]. What is the measure of the diagonal set {(x,x): x \in [0, 1]}? The repeated integrals are unequal, but how do we define its measure? What do we say in this case?

Query: Is an L^1 function defined on R^2 can be approxiated with elemetary function(finite disjoint union of measurable rectangles)?