Today we covered Tao 8.3, 8.4, and 8.5. Here is the video, but I made a stupid mistake regarding Fubini theorem.

I made a mistake in today's presentation in 8.5. Namely, given a measurable function $f(x,y)$. First of all, for a fixed $x$, the function $f_x(y) = f(x,y)$ as a function of $y$, may not be measurable at all. For example, take a measurable subset $E \In \R^2$, it is possible that certain slice $E_x = E \cap \{x\} \times \R$, when viewed as a subset of $\R$, is non-measurable (it is measurable as a subset of $\R^2$, a null-set), then consider $f$ as indicator function $1_E(x,y)$. Hence, the proper way to state the Fubini theorem, is that, there exists a measurable function $F(x)$, such that there exists a null-set $Z$, and for $x \notin Z$, we have $f_x(y)$ is measurable, and $$ F(x) = \int f_x(y) dy. $$ and $$ \int F(x) dx = \int f(x,y) dx dy $$

I will revisit this theorem on Thursday.