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.
Attached is an excerpt from the answer received on this post:
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 $x \mapsto 1-x$.
I know very little about this type of measure, and hence have had trouble seeing why the above is easy to show. I tried to ask the responder about this point, but to no avail. Hence, I am posting this to see if anyone visiting would be up for either explaining this point about invariance, or giving me a push in the right direction to proving it.
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.
Thank you in advance for any response!