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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2015 • Question 5

Question

Math 6210 • Real Analysis Fall 2015 • Question 5

(a) Let \(A\subset [0,1]\) and \(A\in \mathcal B\). Show that for all \(\epsilon >0\) there exists a finite number of intervals \(I_1,\ldots ,I_n\) so that \[ \lambda \left (A\Delta \left (\cup _{j=1}^n I_j\right )\right )<\epsilon . \] Note: \(\Delta \) denotes symmetric difference. So \[ A\Delta B=\{x\in A\cup B:x\notin A\cap B\}. \]

(b) Show that we cannot assume that \(A\subset \cup _{j=1}^n I_j\).

Complex Analysis

Let \(\mathbb D\) denote the open unit disk in \(\mathbb C\). A region is an open connected subset of \(\mathbb C\).

Raw LaTeX

(a) Let \(A\subset[0,1]\) and \(A\in\mathcal B\). Show that for all
\(\epsilon>0\) there exists a finite number of intervals \(I_1,\ldots,I_n\) so
that
\[
\lambda\left(A\Delta\left(\cup_{j=1}^n I_j\right)\right)<\epsilon.
\]
Note: \(\Delta\) denotes symmetric difference. So
\[
A\Delta B=\{x\in A\cup B:x\notin A\cap B\}.
\]

(b) Show that we cannot assume that \(A\subset\cup_{j=1}^n I_j\).

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\textbf{Complex Analysis}

Let \(\mathbb D\) denote the open unit disk in \(\mathbb C\). A region is an open
connected subset of \(\mathbb C\).

Topic Tags

Fourier Analysis and Convolution Riemann Mapping Theorem and Simply Connected Domains