I am studying product measures (From Real Analysis, Royden), basically the application of Fubini and Tonelli theorems.
As much as I understand the theorems, I find it difficult doing computation with the theorems because I am finding it difficult to grasp some of the concepts.
I don't know if it is acceptable here, I have an old notes with a specific problem that I need help understanding, a picture is posted below.
Concern 1: Why is it that when we fix $x$, we have $f(x,y)=0$ a.e [I am thinking it is so because if we fix $x$ then there is only one possible value of $y$, $y=x$ that gives us f(x,y)=1] and so $\int_Xf(x,y)d\mu=0$ But when we fix a $y$ it says $f(x,x)=1$ and $\int_Yf(x,y)d\nu=1$ ? I know the $\sigma$-finiteness condition is violated by the counting measure. but I dont know why there should be any difference in $f$ between when $x$ is fixed and when $y$ is fixed. I was thinking it should be same.
Could someone help clarify that for me, kindly add any other relevant information I may be misssing. And please pardon me if the question doesnt seem well structured. I am new to the product measure concept. Bu I will be glad to clarify if anything is unclear. thanks.
For fixed $x$, $f(x,y)=1$ only when $x=y$. But $\mu(\{x\})=0$ so that $f(x,y)=0$ $\mu$-a.e. and $\int_X f(x,y)d\mu(x)=0$. On the other hand, when $y$ is fixed $\int_Y f(x,y)d\nu(y)=1\times\nu(\{y\})=1$.