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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?

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    @QiaochuYuan I just translated the german term for it word by word and hoped people would understand...2012-12-28

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