I am reading Teller and Fine's article A Characterization of Conditional Probability and I am having trouble checking one of the claims.
A probability space $(X, \mathcal{F}, P)$ is full if for every $A\in\mathcal{F}$, $P$ is a surjection from $\{B\in\mathcal{F}:B\subset A\}$ onto $[0, P(A)]$. The claim is that every probability space can be embedded into a full probability space (an embedding of $(X, \mathcal{F}, P)$ into $(X',\mathcal{F}', P')$ in this context is an injective homomorphism $f: \mathcal{F}\rightarrow\mathcal{F}'$ such that $P'f=P$). The intuitive construction is to embed $(X, \mathcal{F}, P)$ into $(X\times[0,1], \mathcal{F}\otimes\mathcal{B}, P\otimes\mu)$ where $\mathcal{B}$ is the Borel algebra and $\mu$ the uniform probability distribution (the embedding map is $A\mapsto A\times[0,1]$). However, I am having a hard time seeing why $(X\times[0,1], \mathcal{F}\otimes\mathcal{B}, P\otimes\mu)$ is full. Clearly all the measurable rectangles satisfy the desired condition. My current strategy is to check that the collection $\{S\in\mathcal{F}\otimes\mathcal{B}: P:\{T\in\mathcal{F}: T\subset S\}\rightarrow[0, P(S)]\text{ is a surjection}\}$ is a $\sigma$-field. But I got stuck here...any suggestion would be greatly appreciated.