5
$\begingroup$

Possible Duplicate:
Hyperreal measure?

In every proof that the measure problem is unsolvable, the following assumption is made: $$\sum_{k=0}^\infty a=\begin{cases}0\quad a=0\\\infty \;\;a>0\end{cases}$$ But what if we instead define a measure $\mu:X\rightarrow *\mathbb{R}$ (Instead of $\mu:X\rightarrow\mathbb{R}$)

For NSA numbers, identities like $\omega\frac{a}{\omega}=a$ are well-defined even for infinite $\omega$; Therefore the examples and constructions of non-measurable sets (auch as Vitali etc) using the Axiom of Choice become invalid; Are there constructions for sets that avoid the validity of measures in NSA? Is there any further literature about this that is comprehensible for a undergraduate?

  • 2
    This is similar to: http://math.stackexchange.com/questions/257655/hyperreal-measure2012-12-28
  • 0
    @IsaacSolomon I guess that answers my question. Thank you!2012-12-28
  • 1
    What is the measure problem?2012-12-28
  • 0
    @QiaochuYuan the issue that there can be no measure that is defined for every subset of $\mathbb{R}^n$. I am not sure if measure problem is the correct term for it?2012-12-28
  • 0
    I didn't see any references calling that problem by that name after googling variants of "measure problem."2012-12-28
  • 0
    @QiaochuYuan I just translated the german term for it word by word and hoped people would understand...2012-12-28

0 Answers 0