Lebesgue measure of the graph of a function - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T01:21:14Z http://mathoverflow.net/feeds/question/63284 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63284/lebesgue-measure-of-the-graph-of-a-function Lebesgue measure of the graph of a function Cosmonut 2011-04-28T11:09:30Z 2011-04-28T12:06:14Z <p>Let $f: R^n \rightarrow R^m$ be any function. Will the graph of f always have Lebesgue measure zero ?</p> <p>1) I could prove that this is true if f is continuous.</p> <p>2) I suspect it is true if f is measurable, but I'm not sure. (My idea was to use Fubini's theorem to integrate the indicator function of the graph, but I don't know if I'm using the theorem properly).</p> <p>If 2) is incorrect, what would be a counterexample where the graph of f has positive measure ?</p> <p>If 2) is correct, can we prove the existence of a non-measurable function whose graph has positive measure ?</p> http://mathoverflow.net/questions/63284/lebesgue-measure-of-the-graph-of-a-function/63290#63290 Answer by Stefan Geschke for Lebesgue measure of the graph of a function Stefan Geschke 2011-04-28T12:06:14Z 2011-04-28T12:06:14Z <p>If $f$ is a measurable function, then its graph is a measurable subset of $\mathbb R^{n+m}$. By Fubini's theorem, the measure of the graph is 0.</p>