Math 6210 • Real Analysis

Let \((X, \mathcal{M}, \mu)\) be a measure space and \(f : X \to Y\) be a function.
\begin{enumerate}

\item
Show that \(\mathcal{N} = \{f^{-1}A : A \in \mathcal{M}\}\) is a \(\sigma\)-algebra.


\item
Let \(\nu(B) = \mu(f(B))\) for \(B \in \mathcal{N}\). Show that \(\nu\) is a measure.


\item
Show that even if \((X, \mathcal{M}, \mu)\) is complete we do not necessarily have
that \((Y, \mathcal{N}, \nu)\) is complete.

\end{enumerate}