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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2021 • Question 2

Question

Math 6210 • Real Analysis Fall 2021 • Question 2

Let \((X, \mathcal {M}, \mu )\) be a measure space with \(\mu (X) = 1\) and let \(E_{1}, \ldots , E_{n}\) be measurable subsets of \(X\). Suppose that every \(x \in X\) belongs to at least \(k\) of the sets \(E_{i}\). Show that at least one of the sets \(E_{i}\) has measure

\[ \mu (E_{i}) \ge \frac {k}{n}. \]

Raw LaTeX

Let \((X, \mathcal{M}, \mu)\) be a measure space with \(\mu(X) = 1\) and let \(E_{1}, \ldots, E_{n}\)
be measurable subsets of \(X\). Suppose that every \(x \in X\) belongs to at
least \(k\) of the sets \(E_{i}\). Show that at least one of the sets \(E_{i}\) has measure

\[
\mu(E_{i}) \ge \frac{k}{n}.
\]

Topic Tags

Absolute Continuity and Singular Measures Measurable Functions Measure Theory Outer Measures and Caratheodory Measurability