Week 4 Worksheet

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1. Check that $$\mu (B\mathrm{△}C)=\int |1_B-1_C|\mathsf{d}\mu$$ and give an (unnecessarily complicated) proof of the triangle inequality for the quantity $$\rho (B,C)=\mu (B\mathrm{△}C)$$ 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 $(X,\mathcal{P}(X))$. The Lebesgue space ${\mathsf{L}}^{\mathsf{p}}(X,\mathcal{P}(X),\mu )$ is often denoted simply ${\ell }^{\mathsf{p}}(X)$.
   1. Fix $N\in \mathbb{N}$. 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 $f:\mathbb{N}\to [0,\mathrm{\infty })$ 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 $1\le p<q$.
4. Fix a measure space $(X,\mathcal{B},\mu )$ with $\mu (X)<\mathrm{\infty }$. Fix $p\ge 1$. Verify that if $f$ belongs to ${\mathsf{L}}^{\mathsf{p}}(X,\mathcal{B},\mu )$ then $f$ belongs to ${\mathsf{L}}^{\mathsf{q}}(X,\mathcal{B},\mu )$ 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 $b:\mathbb{N}\to \mathbb{R}$ define $T(b):\mathbb{N}\to \mathbb{R}$ by $(Tb)(n)=b(n+1)$.
   1. Fix $p\ge 1$. Check that if $b$ belongs to ${\ell }^{\mathsf{p}}(\mathbb{N})$ then $T(b)$ belongs to ${\ell }^{\mathsf{p}}(\mathbb{N})$.
   2. Is $T$ an isometry of the normed vector space ${\ell }^{\mathsf{p}}(\mathbb{N})$?
7. Fix $n\in \mathbb{Z}$ and let $I_n=[n,n+1]$. Use the Riemann integral on $I_n$ to define a functional on $\mathsf{C}(I_n)$. Verify it is bounded and linear. Apply the Riesz representation theorem to get a Borel measure ${\mu }_n$ on $I_n$. Prove that $$\sum _{n\in \mathbb{Z}}{\mu }_n$$ is equal to Lebesgue measure.