6210-2023-01

Let \((B,\|\cdot \|)\) be a Banach space. Explain the inclusion of \(B\) into its double dual. Prove that it is an isometry onto the image.

Let \(X\) be a set and \(\mathcal {A}\) be an algebra on \(X\). Let \(\rho _0:\mathcal {A}\to [0,\infty ]\) be a pre-measure on \(\mathcal {A}\). Recall that \[ \rho ^*(S)=\inf \left \{\sum _{i=1}^{\infty }\rho _0(A_i): A_i\in \mathcal {A}\text { for all }i\text { and }S\subset \bigcup _{i=1}^{\infty }A_i\right \} \]

is an outer measure. Define what it means that \(\rho _0\) is a pre-measure and show that if \(A\in \mathcal {A}\) then \(\rho ^*(A)=\rho _0(A)\).

Let \((X,\mathcal {M},\mu )\) and \((Y,\mathcal {N},\nu )\) be two measure spaces and \(A\) be measurable with respect to \(\mathcal {M}\times \mathcal {N}\), the \(\sigma \)-algebra generated by measurable rectangles. Show that if \(A_1\subset A_2\subset \cdots \) satisfy that \[ (\mu \otimes \nu )(A_i)=\int _X \nu (\{y:(x,y)\in A_i\})\,d\mu (x) \]

for all \(i\) then

Let \((X,\mathcal {M},\mu )\) be a measure space. Give an example of \(f_1,\ldots :X\to \mathbb {C}\), measurable, which converge in measure to the zero function but do not converge \(\mu \)-a.e. Show that there exists a subsequence \(n_1,n_2,\ldots \) so that \(f_{n_1},\ldots \) converges almost everywhere to the zero function.

In this problem, let \(\lambda \) denote Lebesgue measure on \(\mathbb {R}\) and \(\widehat {\cdot }\) denote the Fourier transform (on \(\mathbb {R}\)). Show that if \(f,g\) are Lebesgue integrable and \(\hat f=\hat g\) then \(f=g\) \(\lambda \)-a.e.

Assume that \(\mu :\mathcal {B}_{\mathbb {R}^{4}}\to [0,\infty ]\) is a measure which is outer regular and \(\sigma \)-finite. Show that if \(\mu \) is inner regular on open sets then it is inner regular on all Borel sets. Recall that a Borel measure on \(\mathbb {R}^{4}\) is called outer regular if for any \(A\in \mathcal {B}_{\mathbb {R}^{4}}\) we have \[ \mu (A)=\inf \{\mu (U): U\text { is open and }A\subset U\} \]

and it is called inner regular if