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\textbf{MATH 501, Analysis, Homework 7}
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\underline{\textbf{1. (2 pts)}} Let $\mu$ and $\nu$ be signed measures on the measurable space $(\X,\M)$. Prove that $\mu \perp \nu$ if and only if $\mu \perp \nu^+$ and $\mu \perp \nu^-$.\\

\underline{\textbf{2. (3 pts)}} Let $(\X,\M,\mu)$ be a measure space and let $f: \X \rightarrow \mathbb{R}$ be in $L^1(\X,\M,\mu)$. Consider the measure defined by $\nu(E)=\int_E f\ d\mu$ for all $E \in \M$. Find the Hahn decomposition of the signed measure space $(\X,\M,\nu)$ and the Jordan decomposition of $\nu$ in terms of $f$ and $\mu$.\\

\underline{\textbf{3. (3 pts)}} Let $\mu: \mathcal{P}(\mathbb{N}) \rightarrow [0,\infty]$ and $\nu: \mathcal{P}(\mathbb{N}) \rightarrow [0,\infty]$ be the measures on the measurable space $(\mathbb{N},\mathcal{P}(\mathbb{N}))$ given by
\[ \mu(A)=\sum_{n \in A} n^2\]
and
\[ \nu(A)=\sum_{n \in A} 2n\]
Show that $\mu \ll \nu$ and find the Radon-Nikodym derivative $\displaystyle \frac{d\mu}{d\nu}$.\\


\underline{\textbf{4. (2 pts)}} Consider the measures $\m$ and $\mu$ on the measurable space $([0,1],\B([0,1]))$, where $\m$ is the Lebesgue measure and $\mu$ is the counting measure. Prove that $\m \ll \mu$ and there does not exists a measurable function $f: [0,1] \rightarrow [0,\infty)$ such that $\m(A)=\int_{A} f\ d\mu$.
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\textbf{Solutions}
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