# Integration along fibres of continuous map on compact Hausdorff spaces

Let $$p:Z\to X$$ be a continuous surjective map between compact Hausdorff spaces. Does there exist a family $$m=(m_x)_{x\in X}$$ of Radon probability measures on $$Z$$, such that

• the support of $$m_x$$ is contained in the fibre $$p^{-1}(\{ x\})$$,

• for every $$f\in C(Z)$$ we have $$f^m\in C(X)$$, where $$f^m(x)=\int_Zf(z)\ dm_x(z)$$.

If not, under which additional restrictions does it exist? I am particularly interested in the case, when $$Z$$ is not first countable.

• Suppose that $p : Z \to X$ is locally trivial with typical fibre $F$ and that $\mu$ is a Radon probability measure on $F$, such that transition functions of $p : Z \to X$ can be taken to be $\mu$-preserving. One should then be able to use a suitable "atlas of local trivialisations" together with a subordinate partition of unity to paste together such a family $m$? If $p : Z \to X$ is a principal $G$-bundle with $G$ a compact group and $\mu$ is the Haar measure on $G$, then this should recover the usual integration along the fibres unless I'm missing something embarrassingly obvious. Mar 30 at 19:45

I think that such family might not exist even for rather well-behaved spaces. Consider $$Z = [0,3]$$ and $$X = [0,2]$$. Then, let $$p \colon Z \to X$$ be defined by: $$p(z) = \begin{cases} z, &\text{ if } z \in [0,1], \\ 1, &\text{ if } z \in [1,2], \\ z-1, &\text{ if } z \in [2,3]. \end{cases}$$ For $$x \in X \setminus \{1\}$$ measure $$m_x$$ has to be Dirac's delta $$\delta_{p^{-1}(x)}$$ as fibres $$p^{-1}(x)$$ are singletons for such $$x$$. Now, let $$f \in C(Z)$$ be any such that $$f \equiv 0$$ on $$[0,1]$$ and $$f\equiv 1$$ on $$[2,3]$$. Then $$f^m(x) = \int_Z f \ \mathrm{d}m_x(z) = \begin{cases} 0, &\text{ for } x \in [0,1), \\ \text{something}, &\text{ for } x =1, \\ 1, &\text{ for } x \in (1,2]. \end{cases}$$ Such $$f^m$$ cannot be continuous regardless of the value of measure $$m_1$$.