# a doubt on transformation on space of probability measure

Suppose, $$X$$ is some measurable space, $$\mu$$ is a probability measure on it, could anyone explain to me how the following is a map or transformation on $$\mathcal M_1(X)$$ (space of all probability measure on $$X$$)? Also, in this case, what is $$T$$ on Suppose, $$X$$ is some measurable space, $$\mu$$ is a probability measure on it, could anyone explain to me how the following is a map or transformation on $$\mathcal M_1(X)$$ (space of all probability measure on $$X$$)?

$$T(\mu)(A)=\mu(T^{-1}(A)$$? now my doubt is $$T$$ acting as a function on $$\mathcal M_1(X)$$, which one is varying here as an input? $$A$$ or $$\mu$$? or both? where $$A\in B(X)$$, $$B(X)$$ is all Borel sets of $$X$$. Thanks! ?

Also, what is the transformation $$T$$ on $$\mathcal M_1(X)$$ , in this case? $$T(A)=\mu(T^{-1}(A)$$?

$$T$$ is assumed to be a measurable transformation in the sense $$T^{-1}(A)$$ is a measurable set whenever $$A$$ is a measurable set. Given any probability measure $$\mu$$ you define a new probability measure $$\nu$$ by $$\nu(A) =\mu (T^{-1}(A))$$. The induced transformation is defined by $$T_*(\mu)=\nu$$. It maps probability measures to probability measures.
• now from this definition of $\nu$ what is the definition of invariant probability measures? $\nu(A) =\mu (T^{-1}(A))=\mu(A)\forall A$, something like this? – miosaki Dec 3 at 0:19
• @miosaki Exactly. Invariance means $\mu =\nu$. – Kabo Murphy Dec 3 at 0:21
• I stil dont understand the role of $T_*$ in the discussion above screenshot. – miosaki Dec 3 at 0:25
• did you mean $T_*$ as an induced transformation? – miosaki Dec 3 at 0:34
• @miosaki By an abuse of notation they are writing $T$ for $T_{*}$ also. – Kabo Murphy Dec 3 at 0:36