Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu\_f, \mu\_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (signed) measures.


Let further $h:= f\star g : [0,1]\rightarrow\mathbb{R}$ be the [$x=1/2$ centered] concatenation of $f$ and $g$ (that is: $h(x) = f(2x)$ for $x\in[0,1/2]$, and $h(x)= g(2x-1) + (f(1)-g(0))$ for $x\in(1/2,1]$).


Is there an algebraic (or otherwise explicit or insightful) relation between the measures $\mu\_h$ and $\mu\_f, \mu\_g$?



Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation?
$\newcommand{\B}{\mathcal B}\renewcommand{\S}{\mathcal S}$Note that for any $a$ and $b$ such that $0\le a\le b\le1$
\begin{equation}
 \mu\_h((a,b])=h(b)-h(a)=
 \left\{
 \begin{alignedat}{2}
& \mu\_f(2(a,b])&&\text{ if }(a,b]\subseteq[0,1/2], \\ 
& \mu\_g(2(a,b]-1)&&\text{ if }(a,b]\subseteq(1/2,1],
 \end{alignedat}
 \right.
\end{equation}
where $2A:=\{2x\colon x\in A\}$ and $2A-1:=\{2x-1\colon x\in A\}$ for any $A\subseteq\mathbb R$.
So, for any left-open subinterval $(a,b]$ of the interval $[0,1]$,
\begin{equation}
\begin{aligned}
 \mu\_h((a,b])&= \mu\_h((a,b]\cap[0,1/2])+\mu\_h((a,b]\cap(1/2,1]) \\ 
 &=\tilde\mu\_h((a,b]) \\ 
 &:=\mu\_f(2((a,b]\cap[0,1/2]))+\mu\_g(2((a,b]\cap(1/2,1])-1). 
\end{aligned} 
\end{equation}
The function $\tilde\mu\_h$ is a finite signed measure on the semiring (say $\S$) of all left-open subintervals of the interval $[0,1]$, and $\sigma(\S)=\B([0,1])$. So, by the uniqueness of the measure extension (cf. e.g. [Proposition 13 of Kisil - Introduction to functional analysis](http://www1.maths.leeds.ac.uk/%7Ekisilv/courses/math3263m014.html)),
\begin{equation}
\begin{aligned}
\mu\_h(B)&=\mu\_f(2(B\cap[0,1/2]))+\mu\_g(2(B\cap(1/2,1])-1) \\ 
&=\mu\_f(2B\cap[0,1])+\mu\_g((2B-1)\cap(0,1]) 
\end{aligned}
\end{equation}
for all $B\in\B([0,1])$.


