On Measure Invariance for a 2-valued Transformation

We consider a family S=S(a) of 2-valued transformations of special form on the segment [0,1] with measure $\mu =\int p(x)d\lambda$, which is absolutely continuous with respect to the Lebesgue measure $\lambda$. 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.