Let $(X,B,\mu)$ and $(Y,C,\nu)$ be probability spaces, and let $m$ be the product measure. Let $f:X \times Y \rightarrow [0,\infty )$ be a $B \otimes C $ measurable function, $1 < p < \infty$, and define $f\_x(y):=f(x,y)$.


What is the most general condition on $f$ to make sure that: $||f||\_{L\_p(m)}=\int{||f\_x||\_{L\_p(\nu)}}d\mu(x)$ ?


I know that if $\int{|f\_x(y)|^p}d\nu(y)$ is constant there is equality. But is it the only case?



Probability measure product space
yes, it is the only case. Denote $g(x)=(\int{|f(x,y)|^pd\nu(y))}^{1/p}$, then your condition is $||g||\_1=||g||\_p$ which is known to imply that $g$ is a constant function, see [here](http://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality).


