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I am trying to prove that Jordan measures satisfy with the following properties $A, B \subset \mathbb R$ and $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$, similarly for $B$:

$$\bar{\mu} (A) \leq \bar{\mu} (A \cup B) \leq \bar{\mu}(A) + \bar{\mu}(B)$$

I am uncertain of the terminology here. So I want to know why there is a restriction $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$? How does it change the problem without and with the assumption? What kind of things we have if we do not have the assumption about the limit?

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    I think that the restriction just say that your sets $A$ and $B$ are bounded. This is because the Jordan measure is only defined for bounded sets. See http://en.wikipedia.org/wiki/Jordan_measure2011-09-09
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    Well, if either A or B are $\infty$, then the inequality will always become trivial, with $\infty$ in each term. If either of the sets is infinite, it can only be covered by the whole of $\mathbb R$, whose measure is $\infty$2011-09-10
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    hhh: BTW, I answered the question in the other post; let me know if you have more questions.2011-09-10
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    @gary: sorry but this question plays with uncountable sets, the latter question played with countable sets [here](http://math.stackexchange.com/questions/63155/jordan-measures-open-sets-closed-sets-and-semi-closed-sets/63184#63184).2011-09-10
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    But the answer I gave below does not assume the sets A,B are countably infinite. I am not sure I get your point.2011-09-11

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