What is the $\sigma$-algebra generated by a Bernoulli Random Variable? Does this depend on the context and how we define the sample space?
I know that the $\sigma$-algebra generated by the indicator function on event $A$ is always by $\{\phi, A, A^{c}, \mathbb{R}\}$, but was wondering if there is also a general result for Bernoulli.
Context: A question in actuarial science asked for the mean of $X = BY$, where $B\sim\text{Bernoulli}(p)$ and $Y$ is some real-valued variable with a given distribution. To calculate $\mathbb{E}[X]$, we needed to use the tower property to deduce $$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X|B]]=p\mathbb{E}[Y].$$
However, the tower property can only be applied if the $\sigma$-algebra that $Y$ is defined on contains the $\sigma$-algebra generated by $B$. I guess since this is not a math subject these technical details can be glossed over, but I am still curious as to whether a general result exists.
Best Answer
"However, the tower property can only be applied if the $\sigma$-algebra that $Y$ is defined on contains the $\sigma$-algebra generated by $B$". This is not true in your example (assuming that $B$ and $Y$ are independent) where a special case of the tower property (the law of total expectation) is applied to $\mathcal{G}_2=\sigma(B)$ and $\mathcal{G}_1=\{\emptyset,\Omega\}$ (as defined in this section).
Also $\sigma(B)$ is of the form $\{\emptyset,A,A^c,\Omega\}$ with $A=B^{-1}(\{0\})$, which depends on the underlying probability space $(\Omega,\mathcal{F},\mathsf{P})$.