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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1995 • Question 5

Question

Math 6210 • Real Analysis Fall 1995 • Question 5
Let \(f\) be an increasing function on \([a,b]\subset\mathbb R\). Use the fact that there is a Borel measure \(\mu\) on \([a,b]\) such that \(\mu([a,x])=f(x)-f(a)\) for all \(a\le x\le b\) to prove that \[ \int_a^b f'\,dm\le f(b)-f(a), \] where \(m\) denotes Lebesgue measure on \([a,b]\).

Raw LaTeX

Let \(f\) be an increasing function on \([a,b]\subset\mathbb R\). Use
the fact that there is a Borel measure \(\mu\) on \([a,b]\) such that
\(\mu([a,x])=f(x)-f(a)\) for all \(a\le x\le b\) to prove that
\[
\int_a^b f'\,dm\le f(b)-f(a),
\]
where \(m\) denotes Lebesgue measure on \([a,b]\).

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Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Lebesgue Measure and Translation Invariance Measure Theory