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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1993 • Question 7

Question

Math 6410 • ODE Fall 1993 • Question 7

Let \(f:G\longrightarrow \mathbb {R}\) be a measurable function such that \(\int _{G_0}\lvert f\rvert \,dx<\infty \) where \(G_0\) is an open subset of \(G\) with \(\overline {G_0}\) compact, i.e. \(f\in L^1_{\operatorname {loc}}(G)\), show that each such \(f\) defines a unique distribution on \(G\). Give an example of a distribution which is not defined by such an \(f\).

Raw LaTeX

Let \(f:G\longrightarrow\R\) be a measurable function such that
\(\int_{G_0}\lvert f\rvert\,dx<\infty\) where \(G_0\) is an open subset of \(G\) with
\(\overline{G_0}\) compact, i.e. \(f\in L^1_{\operatorname{loc}}(G)\), show that each
such \(f\) defines a unique distribution on \(G\). Give an example of a distribution
which is not defined by such an \(f\).

Topic Tags

Distributions and Weak Solutions