I need a little help on a homework exercise, and I think you can help me:
Suppose $(\Omega,F,\mathbb{P})$ is a probability space, and $Z:\Omega\to\mathbb{R}$ is an RV s.th. $Z(\omega)> 0 \ \forall\ \omega$, and $E[Z]=1$. Define for any $A\in F: \\Q[A]=E[Z\cdot1_{A}]$.
Show that this is a probability measure, which I did, and then: Show that $Q$ and $\mathbb{P}$ are equivalent. But how can I show this if $\mathbb{P}$ has been nowhere defined?
If I write
"$Q[A]=E[Z\cdot 1_A] =0\Rightarrow P[A]=0$ by positivity of $Z$"
and
"$P[A]=0\Rightarrow 1_A\equiv0\Rightarrow E[Z\cdot1_A]=0=Q[A]$",
is that already enough?!