Let λ and μ be the lebesgue measure and counting measure, respectfully on ( R...
Let and be the lebesgue measure and counting measure, respectfully on . Show that it holds that for
to further conclude that can't be -finite
I know that if we have two -finite measure spaces then their product measure is unique. Fubini then tells us that if the function f is integrable over the product measure then we can interchange the the order of integration. This is not the case.
I realize that the point is in that the counting measure: The counting measure does exactly what it says: it counts the number of elements in a set. So any infinite set has counting measure , while the measure of any finite set is its cardinality. A would have measure wrt. the counting measure. The counting measure is just summation. The Lebesgue integral of the indicator would just be the Lebesgue measure of A.
How do I actually show that the double integrals hold?