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I'm taking a real analysis course and the textbook we use is Real Mathematical Analysis by Pugh. In page 453, exercise 25 (c), it is explicitly stated that I'm supposed to go to this exact site to see that there is a non measurable function $f:[a,b]\rightarrow[0,\infty)$ with non-measurable graph. The problem is, I cannot find any post on this here. Perhaps they have been closed or deleted?

In any case, if a graph of a function $f$ is non-measurable, then $f$ must be discontinuous everywhere, and in such a terrible way so that its graph is non-measurable. I have no clue what kind of function might that be, and I was specifically told to refer to stackexchange.

Divide1918
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  • Maybe this? – David Mitra Dec 20 '20 at 15:51
  • @DavidMitra I don't quite understand the answer in that post. It says that no graph of function can have positive measure, then continues to show that there is a function whose graph has positive measure. Does that mean the function constructed in the answer is actually measurable? – Divide1918 Dec 20 '20 at 15:55
  • The constructed function has positive outer measure. (And the measure of a Lebesgue measurable set is its outer measure.) – David Mitra Dec 20 '20 at 15:57
  • So does it mean the constructed function is in fact non-measurable? – Divide1918 Dec 20 '20 at 15:59
  • Yes, as mentioned at the start, no function can have graph with positive (Lebesgue) measure. – David Mitra Dec 20 '20 at 16:02

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