Hi guys I got a doubt about how to interpret the following:
Given measurable space $(\Omega, \Im ) \hspace{.1cm} \text{and} \hspace{.1cm}(\mathbb{P}_n)_{n\geq0 }$ succession of probability measures defined on $(\Omega, \Im) $ and $\forall \hspace{.1cm} A \in\Im:$ $\mathbb{P}(A) = \sum_{n=0}^{\infty } a_{n}\mathbb{P}_n(A)$
Now if $\hspace{.01cm}$ $\varphi$ is an integrable function with respect to $\mathbb{P}$ $\hspace{.01cm}$, then $\hspace{.01cm}$ $\varphi$ is an integrable function with respect to $\mathbb{P}_n$
I can barely understand why this happens, but i get confused why the reciprocal it's not true:
If $\hspace{.01cm}$ $\varphi$ is an integrable function with respect to $\mathbb{P}_n$, $\hspace{.01cm}$ then$\hspace{.01cm}$ $\varphi$ is an integrable function with respect to $\mathbb{P}$
Why this is false?
To see the failure of the converse, consider $a_n=2^{-(n+1)}$ and $\mathbb{P}_n$ concentrated on the number $3^n$ with $\varphi(x)=x$.