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All:

Say $f$ is a measurable (integrable, actually) function over the Lebesgue-measurable set $S$, with $m(S)>0$.

Now, since $m(S)>0$, there exists a non-measurable subset $S'$ of $S$, and we can then write:

$$S=S'\cup (S\setminus S').$$

How would we then go about dealing with this (sorry, I don't know how to Tex an integral)

$$\int_S f\,d\mu=\int_{S'} f\,d\mu+ \int_{S\setminus S'}f\,d\mu?$$ (given that $S'$ and $S\setminus S'$ are clearly disjoint)

Doesn't this imply that the integral over the non-measurable subset S' can be defined?

It also seems , using inner- and outer- measure, that if $S'$ is non-measurable, i.e. $m^*, neither is $S\setminus S'$.

So I'm confused here. Thanks for any comments.

Edit: what confuses me here is this:

We start with a set equality $A=B$ (given as $S=S'\cup (S-S')$, so that $A=S$, $B=S-S'$, from which we cannot conclude:

$\int_A f=\int_B f$ , it is as if we had $x=y+z$ , but we cannot then conclude, for any decomposition of $x$, that $f(x)=f(y+z)$.

  • 0
    If $S$ is measurable, and $S'$ is not measurable, then $S-S'$ is not measurable. So both $\int_{S'}fd\mu$ and $\int_{S-S'}fd\mu$ are integrals over nonmeasurable sets. You can't define $\int_{S'}fd\mu$ as $\int_Sfd\mu - \int_{S-S'}fd\mu$, because the last integral is not defined either.2011-12-16
  • 0
    The fact that if $S$ is measurable and $S'$ is not measurable then $S-S'$ is not measurable follows from the fact that the $\sigma$-algebra of measurable sets is closed under under differences: $S'= S-(S-S')$, so if $S$ and $S-S'$ are measurable, then so is $S'$.2011-12-16
  • 0
    I would recommend you to take a look at Banach–Tarski paradox, that give a glimpse of *how* bad non-measurable sets might be: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox2011-12-16
  • 1
    Perhaps it's good to set $f=1$, you'll have measure, not integrals. If $A$ is nonmeasurable subset of measurable $x$, it is true that $\mu(X)=\mu(A \cup (X-A))$, but you cannot change this to $\mu(A)+\mu(X-A)$ since RHS is not defined.2011-12-16

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