On Measure Invariance for a 2-valued Transformation

We consider a family of 2-valued transformations of special form on the interval $[0,1]$ with measure $\mu=\int p(x)\,d\lambda$ which is absolutely continuous with respect to the Lebesgue measure. We endow $S$ with a set of weight functions $\alpha=\{\alpha_1(x),\alpha_2(x)\}$ and find a criterion of measure invariance under the transformation. This criterion relates the three parameters $a$, $p$, $\alpha$ to each other.