(1)Please give a real-valued function $f$ satisfies the set $\{(x,f(x)):x\text{ belongs to }\mathbb{R}\}$ is a second category subset of $\mathbb{R}^2$? (2)Please give a real-valued function f satisfies the set $\{(x,f(x)):x\text{ belongs to }\mathbb{R}\}$ is a non-measurable set in $\mathbb{R}^2$ in the Lebesgue sense?
Existence of Real-Valued Functions Satisfying Certain Properties
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real-analysis
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0but how do you proof the range of f is a second category subset of R2? – 2011-08-04
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0@mathabc: Mike’s function is a surjection: its range is *all* of $\mathbb{R}^2$. – 2011-08-04
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0Whoops, you're right that doesn't work. – 2011-08-04
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0my mistake,what I mean is how to proof the graph of f is a second category subset of R2? – 2011-08-04
1 Answers
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See Gelbaum and Olmsted, Counterexamples in Analysis, Chapter 10, Plane Sets, Example 23, A real-valued function of one real variable whose graph is a nonmeasurable plane set. It's a little too long for me to type out, and it depends on Example 21, A nonmeasurable plane set having at most two points in common with any line. Example 21 takes two full pages in the book.
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0Thank you very much!For(1),do you have any ideas? – 2011-08-04
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0You may use space-filling curves, aka Peano curves. – 2011-08-04
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0For the first, wouldn't using a combination using a nonmeasurable subset of $\mathbb R^2$ help? – 2011-08-04
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0@gary, OP wants the graph of a function, which a Peano curve is not. Also, I don't know what "a combination using a nonmeasurable subset" means. – 2011-08-04
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0I meant somehow using the fact that the characteristic function of a nonmeasurable set is nonmeasurable. – 2011-08-04
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0why does the characteristic function of a nonmeasurable set work? – 2011-08-04
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0I don't think the characteristic function of a nonmeasurable set does work. A nonmeasurable subset of the real line is just a measure zero set when viewed as a subset of the plane. – 2011-08-04
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0I have found the two examples.Thank you very much for all your help. – 2011-08-04