Week 4 Worksheet

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1. Check that \[ \mu(B \triangle C) = \int |1_B - 1_C| \intd \mu \] and give an (unnecessarily complicated) proof of the triangle inequality for the quantity \[ \rho(B,C) = \mu(B \triangle C) \] from Worksheet 2.
2. Let $\nu$ be the counting measure on $\{1,2\}$. Interpret Hölder's inequality and Minkowski's inequality in this context. Draw Euclidean pictures to explain the inequalities.
3. Let $\mu$ be counting measure on $(X,\P(X))$. The Lebesgue space $\mathsf{L}^{\!\mathsf{p}}(X,\mathcal{P}(X),\mu)$ is often denoted simply $\ell^{\mathsf{p}}(X)$.
   1. Fix $N \in \N$. Verify that \[ \ell^\mathsf{p}(\{1,\dots,N\}) = \ell^\mathsf{q}(\{1,\dots,N\}) \] for all $p,q > 0$.
   2. Produce for every $q > p \ge 1$ a non-negative function $f : \N \to [0,\infty)$ such that $f$ belongs to $\ell^q$ but does not belong to $\ell^p$.
   3. Prove that $\ell^p$ is a subset of $\ell^q$ whenever $1 \le p \l q$.
4. Fix a measure space $(X,\B,\mu)$ with $\mu(X) \l \infty$. Fix $p \ge 1$. Verify that if $f$ belongs to $\mathsf{L}^{\!\mathsf{p}}(X,\B,\mu)$ then $f$ belongs to $\mathsf{L}^{\!\mathsf{q}}(X,\B,\mu)$ for all $1 \le q \le p$.
5. Compare the two previous questions. In one situation Lebesgue spaces with $p$ small embed in Lebesgue spaces with $p$ larger, and in another Lebesgue spaces with $p$ large embed in Lebesgue spaces with $p$ smaller. Why is there a difference?
6. Given $b : \N \to \R$ define $T(b) : \N \to \R$ by $(Tb)(n) = b(n+1)$.
   1. Fix $p \ge 1$. Check that if $b$ belongs to $\ell^\mathsf{p}(\N)$ then $T(b)$ belongs to $\ell^\mathsf{p}(\N)$.
   2. Is $T$ an isometry of the normed vector space $\ell^\mathsf{p}(\N)$?
7. Fix $n \in \Z$ and let $I_n = [n,n+1]$. Use the Riemann integral on $I_n$ to define a functional on $\cont(I_n)$. Verify it is bounded and linear. Apply the Riesz representation theorem to get a Borel measure $\mu_n$ on $I_n$. Prove that \[ \sum_{n \in \Z} \mu_n \] is equal to Lebesgue measure.