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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2016 • Question 5

Question

Math 6210 • Real Analysis Fall 2016 • Question 5

For \(f:\mathbb R\to \mathbb R\) and \(y\in \mathbb R\), define \(\tau _y f(x)=f(x-y)\) for \(x\in \mathbb R\). Assume that \(\mathbb R\) is equipped with the Lebesgue measure. For \(f\in L^1(\mathbb R)\), show that the map \(\mathbb R\to L^1(\mathbb R)\) defined by \(y\mapsto \tau _y f\) is uniformly continuous.

Raw LaTeX

For \(f:\mathbb R\to\mathbb R\) and \(y\in\mathbb R\), define
\(\tau_y f(x)=f(x-y)\) for \(x\in\mathbb R\). Assume that \(\mathbb R\) is equipped
with the Lebesgue measure. For \(f\in L^1(\mathbb R)\), show that the map
\(\mathbb R\to L^1(\mathbb R)\) defined by \(y\mapsto\tau_y f\) is uniformly
continuous.

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory