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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2014 • Question 2

Question

Math 6210 • Real Analysis Fall 2014 • Question 2

Let \((X,\mathcal B,\mu )\) be a measure space. \(U\in \mathcal B\) is called an atom if \(\mu (U)>0\) and \(\mu (V)=0\) for all \(V\), a proper subset of \(U\). If \((X,\mathcal B,\mu )\) is \(\sigma \)-finite measure space and \(f:X\to Y\) then \(f_*(\mu )\) has at most countably many atoms.

Raw LaTeX

Let \((X,\mathcal B,\mu)\) be a measure space. \(U\in\mathcal B\) is
called an atom if \(\mu(U)>0\) and \(\mu(V)=0\) for all \(V\), a proper subset of
\(U\). If \((X,\mathcal B,\mu)\) is \(\sigma\)-finite measure space and
\(f:X\to Y\) then \(f_*(\mu)\) has at most countably many atoms.

Topic Tags

Absolute Continuity and Singular Measures Measure Theory Outer Measures and Caratheodory Measurability Product Measures and Fubini-Tonelli Theorems