6210-2023-08

Is \(\{v\in \ell ^2(\mathbb {N}): |v_i|\le 1/\log (i+4)\}\) compact in the norm topology on \(\ell ^2(\mathbb {N})\)? Justify your answer.

Let \(f:\mathbb {R}\to \mathbb {C}\) be Lebesgue measurable. Assume that \(f\) is invariant under translations by rational numbers. Show that \(f\) is constant Lebesgue almost everywhere.

Define the Fourier transform on \(\mathbb {R}\) (do not worry about the choice of normalization). Show that the Fourier transform of an \(L^1(\mathbb {R},\lambda )\) function is in \(C_0(\mathbb {R})\). You may use that the Fourier transform of a continuous compactly supported function is in \(C_0(\mathbb {R})\).

Let \(X=[0,1]\) and let \(\mu \) be a finite Borel measure on \(X\). Prove that \(\mu \) is regular, i.e. that for any Borel set \(A\subseteq X\) the following holds:

1. \(\mu (A)=\sup \{\mu (K)\mid K\subseteq A,\ K\text { compact}\}\), and
2. \(\mu (A)=\inf \{\mu (U)\mid U\supseteq A,\ U\text { open}\}\).

Note: The same statement holds for any compact metrizable space \(X\).

Let \(f\) be Lebesgue integrable on \((0,1)\). For \(x\in (0,1)\) define \[ g(x)=\int _x^1 \frac {f(t)}{t}\,d\lambda . \]

Prove that \(g\) is Lebesgue integrable on \((0,1)\) and that \[ \int _0^1 g(x)\,d\lambda =\int _0^1 f(x)\,d\lambda . \]

Let \((X,\mathcal {B},\mu )\) be a finite measure space and \(f:X\to \mathbb {R}\) be measurable.

1. Show that if \(g_n(x)=(f(x))^n\) and \(\int g_n\,d\mu \) is uniformly bounded for all \(n\) then \(|f(x)|\le 1\) (\(\mu \)-a.e.).
2. Show that \(\int _X g_n\,d\mu \) is independent of \(n\) iff there is \(A\in \mathcal {B}\) so that \(f=\chi _A\) (\(\mu \) a.e.).