$\Omega$- is the interval [0,1], $\mathbb{P}_0$ is Lebesgue measure, $\mathbb{P}_1$ is the probability measure given by $\mathbb{P}_1([a,b])=\int_a^b 2\omega d\mathbb{P}_0(w)$ and $\mathbb{P}_2([a,b])=\int_a^b 6\omega (1-\omega)d\mathbb{P}_0(w)$. I task is :
a) Verify that $\mathbb{P}_0, \mathbb{P}_1$ and $\mathbb{P}_2$ are equivalent measures.
b)Write down the Radon-Nikodym derivatives $\frac{d\mathbb{P}_1}{d\mathbb{P}_0}$ and $\frac{d\mathbb{P}_2}{d\mathbb{P}_0}$
c) Write down what are $\frac{d\mathbb{P}_0}{d\mathbb{P}_1}$ and $\frac{d\mathbb{P}_0}{d\mathbb{P}_2}$
d) Write down what is $\frac{d\mathbb{P}_2}{d\mathbb{P}_1}$
Here is my solution -
The probability measure $\mathbb{P}$ and $\overline {\mathbb{P}_1}$ on $(\Omega,F)$ are equivalent if and only if there is an almost surely random variable $Z$: $\mathbb{E}Z=1$ and $\overline {\mathbb{P}}(A)=\int_A Z\omega d\mathbb{P}(w)$ for all $A \in F$. In this case, if $X$ is an integrable RV, then $\overline{\mathbb{E}}X=\mathbb{E}[XZ]$, where $\overline{\mathbb{E}}X$ is the expectation withrespect to the probability measure $\overline{\mathbb{P}}$. If $Z$ is almost surely strictly positive - ${\mathbb{E}}Y$=$\overline{\mathbb{E}}[X/Z]$ for positive RV $Y$. Finally - for any random variable $X$, $\overline{\mathbb{E}}[X]=\int_\Omega X d\overline{\mathbb{P}} =\int_\Omega XZ d\mathbb{P}$, i.e. $d\overline{\mathbb{P}}=Zd\mathbb{P}$ and $Z=\frac{d\overline{\mathbb{P}}}{d\mathbb{P}}$, where $Z$ - Radon-Nikodym derivative.
a) I assumed that I need to check that $Z_1= 2\omega $ and $Z_2=6\omega (1-\omega)$ are Radon-Nikodym derivatives. $Z_1\ge 0$ on [0,1]. $\int_0^1 2\omega d\mathbb{P}_0$=$\mathbb{E}Z_1=1$
b) $Z_1=\frac{d\mathbb{P}_1}{d\mathbb{P}_0}$ and $Z_2=\frac{d\mathbb{P}_2}{d\mathbb{P}_0}$...
c) The Radon-Nikodym possible measures of all equivalents - $\frac{d\mathbb{P}_0}{d\mathbb{P}_1}=\frac{1}{Z_1}$ and $\frac{d\mathbb{P}_0}{d\mathbb{P}_2}=\frac{1}{Z_2}$
d) $\frac{d\mathbb{P}_2}{d\mathbb{P}_1}=\frac{Z_1}{Z_2}$, $\mathbb{P}_2([a,b])=\int_a^b \frac{d\mathbb{P}_2}{d\mathbb{P}_1}d\mathbb{P}_1$
It seems like something is wrong and I don't really understand how to complete part b) Can somebody help? Many thanks!
Friedrich Philipp, thank you for your comments - I am unable to add some. I am understand the strategy, b) - is a problem, because I am confused with $d\mathbb{P}$, because I know that Lebesgue measure on [0,1] can be defined as $\mathbb{P}[a,b]=b-a$ and Lebesgue integral in general = $\int_\Omega \omega d\mathbb{P}$