# A question about the Lebesgue-Stieltjes measure of the Cantor function

This question is a follow-up to the post "Calculating a Lebesgue integral involving the Cantor Function."

Let $\varphi: [0,1] \rightarrow [0,1]$ be the Cantor (ternary) function, and let $m_\varphi$ be the Lebesgue-Stieltjes measure associated to it.

Notice that $\varphi(1-x) = 1 - \varphi(x)$. From this, it is easy to show that the Cantor measure $m_\varphi$ is invariant under the transformation $x \mapsto 1-x$.
Also: the responder changed the notation in the integral from $dm_\varphi$ to $m_\varphi dx$, and I would be grateful for any clarification as to the relationship of these two notations.