Let $T: [0,1] \rightarrow [0,1]$ be a multiplication by $ \beta >1$ mod $1$. Show that $h(x) d x$ is $T$-invariant where
$$h (x) = \sum_{n \geq 0} \beta^{-n} \chi _{[0,T^n (1)]} (x)$$
($\chi$ is indicator function)
I can do this for $\beta =2$ when the indicator functions are trivially equal to one, and this is a constant times Lebesgue' measure. Don't know how to do the general case.
Edited: there was a mistake,$\chi _{[0,T^n ]}$ should be $\chi _{[0,T^n (1)]}$. I am told that this can be found in some paper of 'Perry' but could not find exact references.
Hint: You must show that $$ \int_0^1\varphi h=\int_0^1(\varphi\circ T)h $$ for any integrable function $\varphi$ (with respect to your measure) or, equivalently, that this happens for all functions $\varphi=\chi_{[0,t]}$ for $t\in[0,1]$. For example, the left-hand side becomes $\int_0^1\chi_{[0,t]} h=\int_0^t h$.
PS: You need to clarify in your source what is meant by $\chi_{[0,T^n]}(x)$, although it could be $\chi_{[0,T^n(x)]}$.