A question about pushforward measures and Peano spaces

Question

Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ such that $f_\# \mu(A)=\nu(A)$ for all $A\in \sigma$?

Keep in mind that if $(X,B)$ and $(Y,C)$ are measurable spaces, $f: X \to Y$ is a measurable map, and $\mu$ is a measure on $B$, then we write $f_\#\mu$ for the push forward measure on $C$, which is defined by $f_\#\mu(A) = \mu(f^{-1}(A))$ for all $A \in C$.

Any insights or sources would help.

Answer 1

In general the answer to this problem is negative: if the measure $\mu$ has connected support and the measure $\nu$ has disconnected support, then for any continuous map $f:P\to P$ the measure $f_\# \mu$ will be supported by a connected subset of $P$ and hence will be not equal to the measure $\nu$.

Let us recall that the support $\mathrm{supp}(\mu)$ of a measure $\mu$ on $P$ is the closed set of all points $x\in P$ such that every neighborhood $O_x$ of $x$ has measure $\mu(O_x)>0$.

On the other hand, if $\mu,\nu$ are nonatomic probability measures on Peano continua $X,Y$ and $\mathrm{supp}(\nu)=Y$, then there exists a continuous map $f:X\to Y$ such that $\nu=f_\# \mu$. The construction of the map $f$ consists of five steps:

1. Construct a continuous map $\varphi:[0,1]\to Y$ such that for every nonempty open set $U\subseteq [0,1]$ the image $\varphi[U]$ has nonepty interior and hence has measure $\nu(\varphi[U])>0$. The construction of $\varphi$ uses a piece of graph theory.

2. Choose any probability measure $\lambda$ on $[0,1]$ such that $\varphi_\#\lambda=\nu$ and observe that $\mathrm{supp}(\lambda)=[0,1]$.

3. Choose a compact zero-dimensional set $C\subseteq X$ such that every nonempty relatively open subset $U\subseteq C$ has measure $\mu(U)>0$. Then $C$ is homeomorphic to the Cantor set and hence admits a continuous surjective function $\psi_0:C_0\to[0,1]$ such that $|\psi_0^{-1}(y)|\le 2$ for every $y\in[0,1]$. Using the Baire Category Theorem, choose a continuous function $\psi:X\to [0,1]$ such that $\psi{\restriction}_C=\psi_0$ and for every $y\in [0,1]$ the preimage $\psi^{-1}(y)$ has measure $\mu(\psi^{-1}(y))=0$. Then $\psi_\#\mu$ is a nonatomic measure on $[0,1]=\mathrm{supp}(\psi_\#\mu)$.

4. Choose a monotone homeomorphism $h:[0,1]\to[0,1]$ such that $h_\#(\psi_\#\mu)=\lambda$.

5. The function $f=\varphi\circ h\circ\psi:X\to Y$ will have the desired property: $f_\#\mu=\nu$.