Is the Given Statement a Consequence of Fubini's Theorem?

[Tatianaoo]

Does anybody know if the following is true?

Let $p>1$. If $f=f(x,y)$ is such that $f\in L^p([a,b]\times [a,b])$, then $f_y(x)\in L^p([a,b])$ for almost all $y\in[a,b]$ and $f_x(y)\in L^p([a,b])$ for almost all $x\in[a,b]$.

Is this a consequence of Fubini's theorem?

 

[Vargo]

Unfortunately, $ don't work. You need to enclose your code with [/itex] and $. The one with the slash goes at the end but I had to put them in reverse order for it to show up (or it would have just treated "and" as my code.$

 

[Vargo]

The answer to your question is yes. g is Lp if and only if |g|^p is L1. So apply Fubini to the function $|f(x,y)|^p$, which is L^1.

 

[Tatianaoo]

Thanks a lot for help!