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\textbf{MATH 501, Analysis, Homework 6}
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\underline{\textbf{1. (2 pts)}} Let $(\X,\M,\mu)$ and $(\Y,\N,\eta)$ be measure spaces with $\N \neq \mathcal{P}(\Y)$ and $\mu(A)=0$ for some non-empty $A \in \M$. Show that the product measure space $(\X \times \Y, \M \otimes \N, \mu \times \eta)$ is not complete.\\


\underline{\textbf{2. (3 pts)}} Consider the measure spaces $([0,1],\B([0,1]),\m)$ and $([0,1],\B([0,1]),\eta)$ where $\m$ is the Lebesgue measure and $\eta$ is the counting measure. Let $$D=\{(x,x) \in [0,1] \times [0,1]: x \in [0,1]\}$$ Prove that 
\[ \int_{[0,1] \times [0,1]} \chi_D(x,y)\ d(\m\times\eta)=\infty\]
\\

\underline{\textbf{3. (2+3 pts)}}\\

a) Consider the measure space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$ where the measure $\mu: \mathcal{P}(\mathbb{N}) \rightarrow [0,\infty]$ is given by
\[ \mu(S)=\displaystyle \sum_{n \in S} \frac{1}{3^n}\] Let $f: \mathbb{N} \rightarrow \mathbb{R}$ be any integrable function. Find a sequence of simple functions $f_n: \mathbb{N} \rightarrow \mathbb{R}$ such that $\displaystyle \lim_{n \rightarrow \infty} f_n(k) = f(k)$ for all $k \in \mathbb{N}$. Then express the integral $$\int_{\mathbb{N}} f(k)\ d\mu$$
in terms of $f(k)$'s.\\


b) Now consider the product space \[(\mathbb{R} \times \mathbb{N}, \mathcal{B}(\mathbb{R}) \otimes \mathcal{P}(\mathbb{N}), \m \times \mu)\] where $m$ is the usual Lebesgue measure on $\mathbb{R}$. You are given that the function $g: \mathbb{R} \times \mathbb{N} \rightarrow \mathbb{R}$ defined by $\displaystyle g(x,k)=\frac{2^k}{1+x^2}$ is measurable. Evaluate the integral
\[ \displaystyle \int_{\mathbb{R} \times \mathbb{N}} \frac{2^k}{1+x^2}\ d(m \times \mu)\]
\textbf{Justify each step} of your calculation by referring to the relevant theorems.
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\textbf{Solutions}
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