On the Existence of the Perron-Frobenius Operator (Radon-Nikodym) For a $\sigma$-finite measure space $(X,\mu)$ and a non-singular transformation $T\colon X\to X$ (i.e. $\mu(A)=0$ iff $\mu(T^{-1}(A))=0$ for all measurable $A\subset X$), my lecture notes give the following definition:

For each $f\in L^{1}(X,\mu)$, denote by $P_{T}f$ the unique element in $L^{1}(X,\mu)$ that satisfies $$\int_{A}P_{T}f \text{d}\mu=\int_{T^{-1}(A)}f \text{d}\mu.$$ The operator $P_{T}\colon L^{1}(X,\mu)\to L^{1}(X,\mu)$ is called the Perron-Frobenius operator.

According to the lecture notes, the existence and uniqueness are a consequence of the Radon-Nikdoym theorem. They reason as follows: Define a measure $\nu$ on $X$ by $$\nu(A):=\int_{T^{-1}(A)}f \ \text{d}\mu.$$ Then integrability of $f$ implies that $\nu$ is finite (and hence $\sigma$-finite). The non-singularity of $T$ implies that $\nu<<\mu$ (absolute continuous w.r.t. $\mu$). So the Radon-Nikodym theorem implies the existence of a unique element $P_{T}f\in L^{1}(X,\mu)$ such that $$\nu(A)=\int_{A}P_{T}f \ \text{d}\mu.$$

First question: Why does this proof of the existence and uniqueness work? It seems that $\nu$ can attain negative values.

In the lecture notes, they give the following exercise: Prove that $P_{T}$ is positive. 

Second question: What do they mean by positive? It is not defined in the lecture notes.


Any real valued measure is a difference of two finite positive measures, so RNT for such measures is true.
An operator $V: L^{1}(X,\mu) \to L^{1}(X,\mu)$ is called pisitve if $Vf \geq $ whenever $f \geq 0$. In this case $f \geq 0$ implies $\int_AP_Tf d\mu \geq 0$ for all $A$ which implies that $P_Tf \geq 0$.
