Show that $\mathbb P ' \approx \mathbb P$ exists with bounded density so that $Y^{1},...,Y^{n}\in L^{1}(\mathbb P^{'})$

Question

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $Y^{1},...,Y^{n}$ be random variables. Show that $\mathbb P ' \approx \mathbb P$ exists with bounded density $Z \in L^{\infty}$ so that $Y^{1},...,Y^{n}\in L^{1}(\mathbb P^{'})$

I am unsure on how to begin, as I would have to explicitly state the measure $\mathbb P'$, and to which I have no further information.

Any ideas on how to begin?

Answer 1

Take $Z= c\frac 1 {1+ \sum\limits_{k=1}^{n} |Y^{(k)}|}$where $c>0$ is chosen so that $P'$ defined by $P'(E)=\int_E ZdP$ is a probability measure.