# Invariant measures

I'm working on a paper in which I have to prove that following measure is $$T$$ invariant.
Notice that $$T$$ is defined the way like this :

\begin{align*} T: [0,1] \longrightarrow [0,1] \\ T(x) = \begin{cases} x+ \dfrac{1}{2} & x \in [0,\dfrac{1}{2}]\\ 2x-1 & x \in (\dfrac{1}{2} , 1] \end{cases} \end{align*}
I'm going to prove that $$\mu$$ is $$T$$ invariant
\begin{align*} \mu = \dfrac{\nu + \nu T}{2} \end{align*}
in which $$\nu$$ is a $$T^{2}$$ invariant measure with density $$h$$ with respect to lebesuge measure $$m$$.

I tried alot :

As you know we have to prove for any $$B$$ in $$\mathcal{B}[0,1]$$ we have : \begin{align*} \mu(B) = \mu(T^{-1}B) \end{align*}
So :
\begin{align*} \mu(B) = \dfrac{\nu(B) + \nu T(B)}{2} \end{align*} And we have : \begin{align*} \mu(T^{-1}B) = \dfrac{\nu T^{-1}(B) + \nu(B)}{2} \end{align*}
So we have to prove that :
\begin{align*} \nu T (B) = \nu T^{-1}(B) \end{align*}
We know that $$\nu T^{-1}(B) = \int_{T^{-1} (B)} h dm$$
I computed this :
\begin{align*} \nu T^{-1}(B) = \dfrac{1}{2} h(\dfrac{x+1}{2}) + h(x - \dfrac{1}{2}) 1 _{[\dfrac{1}{2},1]}(x) \end{align*}

I don't know how compute $$\nu T(B)$$ and when do we use from the fact that $$\nu$$ is $$T^{2}$$ invariant ?
Can we say that since $$\nu$$ is $$T^{2}$$ invariant so : \begin{align*} \nu(B) = \nu(T^{-2}(B)) \end{align*} So we have that : \begin{align*} \nu( T T^{-2}(B)) = \nu (TB) \end{align*}

Since $$\nu T^{-2}=\nu$$, we have $$\nu(T(B))=\nu(T^{-2}(T(B)))=\nu(T^{-1}(B)),$$ and thus $$\mu(T^{-1}(B))=\frac{1}{2}\left[\nu(T^{-1}(B))+\nu(B)\right]=\frac{1}{2}\left[\nu(T(B))+\nu(B)\right]=\mu(B).$$
Think of $$\mu$$ as being the average of $$\nu$$ and $$\nu T$$. Now if $$\nu$$ is $$T^2$$ invariant, this means that $$\nu$$ and $$\nu T^2$$ is the same measure and you should be able to conclude directly that $$\mu$$ is $$T$$-invariant. (You average along the $$T$$ orbit using the periodic behaviour of $$\nu$$.)
Bonus question: If you started with a measure say $$\kappa$$ which is $$T^n$$ invariant for some $$n$$ a fixed natural number. How would you construct a $$T$$-invariant measure from $$\kappa$$?