Let $(X,\mathcal{A},\mu)$ be a measure space

where $X$ is uncountable, $\mathcal{A}=\{E \subset X | E \space or \space E^c \space countable\}$ and

$\mu(E) = \left\{ \begin{array}{lr} 0 & : E \space countable \\ 1 & : E^c \space countable \end{array} \right.$

Describe all of the functions measurable w.r.t. $\mathcal{A}$

Can someone please explain this question to me? I really have no idea where to start.

Would all the functions $f_n(x)$ s.t. $f_n^{-1}(E)$ is countable or $(f_n^{-1}(E))^c$ is countable for any open set $E \in \mathcal{A}$ be a suitable answer?

After describing the functions I must integrate them so I am not sure if my description is sufficient.

  • $\begingroup$ sorry, was a typo $\endgroup$ – Pablo May 22 '14 at 20:25
  • 2
    $\begingroup$ This is just shooting from the hip, but I have a hunch that the functions must be constant except for a countable number of points. Else, for some $a$, $f^{-1}(a, \infty) \bigcup f^{-1}(-\infty, a] = X$ would be a partition of $X$ into two uncountable sets, which is against the assumption. $\endgroup$ – Darrin May 23 '14 at 1:30

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