What is the difference between $\mathcal B(E\times F)$ and $\mathcal B(E)\bigotimes\mathcal B(F)$
I know the definition of $\mathcal B(E)\bigotimes\mathcal B(F)=\sigma\{A\times B:A\in\mathcal B(E), B\in\mathcal B(F)\}$
I heard that, $\mathcal B(E)\bigotimes\mathcal B(F)\subset\mathcal B(E\times F)$ with the following argument
$\pi_1:E\times F\to E$ and $\pi_2:E\times F\to F$ must be continuous, hence measurable. So if we endow these sets with the corresponding $\sigma$-algebras, we get for every $A\in\mathcal B(E),\ \pi_1^{-1}(A)\in\mathcal B(E\times F)$ and the same with the second projection, but what have we reached now, what has this to do with the definition above ?
Do you know an example, where the inclusion is strict, i.e. $\mathcal B(E)\bigotimes\mathcal B(F)\subsetneq\mathcal B(E\times F)$
I think i found it in the lecture notes, but the example is missing
http://imgim.com/img_20140404_143200.jpg (here the proposition, we need for the following lemma)
http://imgim.com/img_20140404_143214.jpg ( and the lemma, there's a typo in the lemma $f_2$ maps to $(F_2,\mathcal B_2)$)
