Suppose we have Lebesgue measurable set E. Let F be a subset of E with measure zero with respect to the Lebesgue measure on E. My question is, can we construct a "reasonable" integration theory on the set F that can be extended to the set E ? In other words, what are the alternative theory of integrations on a set of measure zero (the measure zero sets are with respect to Lebesgue measure ) ?

  • $\begingroup$ I don't understand the question, perhaps you could elaborate more $\endgroup$ – Amr Dec 30 '13 at 17:55
  • $\begingroup$ Measure zeros sets are kind of "bad" sets (here it is F) where Lebesgue measure vanishes, my question is, can we construct a non-vanishing "good" measure on F such that the integration with respect to that measure is non zero on F and with respect to the same hypothetical measure on F, an Integration theory be constructed on the larger set E ? $\endgroup$ – user118248 Dec 30 '13 at 18:00
  • $\begingroup$ So, for example, you're asking what it takes to recover the Lebesgue measure on $\mathbb R$ from the counting measure on $\mathbb Z$? $\endgroup$ – Chris Culter Dec 30 '13 at 18:13
  • $\begingroup$ Partially yes, but I am looking for a "big picture", for say rationals have zero measure inside reals with respect to Lebesgue so, I want an Integration theory, entirely built over rationals and non-zero and will give a well defined meaning over reals, and of course the theory should be generalised in an appropriate sense whenever possible $\endgroup$ – user118248 Dec 30 '13 at 18:17

On many sets of measure zero, you can construct the "Hausdorff measure." The Cantor set is a good example of such a set. http://en.wikipedia.org/wiki/Hausdorff_measure

For countable sets like the rationals, I think measures absolutely continuous with respect to counting measure are all you are going to get.

  • $\begingroup$ Thank you, I do have some knowledge on Hausdorff measure, but I want to have a more general theory than that, as you can't apply "Hausdorff" measures without a metric ! More in some abstract setting so that I can apply the theme for general spaces . $\endgroup$ – user118248 Dec 30 '13 at 20:36
  • $\begingroup$ So on locally compact topological groups, you get Haar measure. And many other structures give rise to measures. But if you have a set with no additional structure, then the only way to categorize the set is by cardinality. Obviously sets like the Cantor set which have the same cardinality of the real numbers can be given any number of measures. In fact the reals can have any number of weird measures placed upon them, simply by taking a non-trivial bijection between $\mathbb R$ and $\mathbb R$ (or $\mathbb R^n$ is you like). $\endgroup$ – Stephen Montgomery-Smith Dec 30 '13 at 20:49
  • $\begingroup$ Suppose you are given an infinite set? How do your make rigorous the notion that any element of the set can be picked with equal likelihood? This is one of the central questions of Bayesian statistics. Really, nobody has much of a clue. $\endgroup$ – Stephen Montgomery-Smith Dec 30 '13 at 20:55
  • $\begingroup$ Thank you for the information regarding Bayesian Statistics, actually I want to define a test function on certain set of measure zero within a larger space, I have a "theory of integration" in that set of measure zeo, but can't extend the theory on the larger set, as if a function h is defined over rationals, but if I integrate with respect to Lebesgue measure of reals, I can never get a non-zero value of the integral of h, so I need a way out, that is why the question $\endgroup$ – user118248 Dec 30 '13 at 21:03
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    $\begingroup$ The idea is to find a set $C$ that contains $A \cup B$ such that $C \setminus (A \cup B)$ has cardinality the same as $A$, and $C$ has measure zero. Then find a bijection $\phi:C \to C$ that takes $A$ to $B$. Then define $f:\mathbb R \to \mathbb R$ as the identity on $\mathbb R \setminus C$, and $\phi$ on $C$. $\endgroup$ – Stephen Montgomery-Smith Dec 30 '13 at 21:39

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