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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2021 • Question 3

Question

Math 6210 • Real Analysis Spring 2021 • Question 3

Let a measure space \((X, \mathcal {M}, \mu )\) be given and for a fixed measurable function \(f : X \to [0, \infty )\) define a new measure \(\lambda \) on \(\mathcal {M}\) by \[ \lambda (E) = \int _{E} f\,d\mu . \]

You don’t have to prove that \(\lambda \) is a measure. Prove that for any measurable \(g : X \to [0, \infty )\), \[ \int g\,d\lambda = \int gf\,d\mu . \]

Raw LaTeX

Let a measure space \((X, \mathcal{M}, \mu)\) be given and for a fixed measurable
function \(f : X \to [0, \infty)\) define a new measure \(\lambda\) on \(\mathcal{M}\) by
\[
\lambda(E) = \int_{E} f\,d\mu.
\]


You don't have to prove that \(\lambda\) is a measure. Prove that for any
measurable \(g : X \to [0, \infty)\),
\[
\int g\,d\lambda = \int gf\,d\mu.
\]

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Measurable Functions Measure Theory