Why we can not extend Lebesgue outer measure for all subsets of real line? I wonder that why we can't extend the Lebesgue outer measure for all the subsets of real line?
 Why we can't define another such measure on subsets of reals?
I can't image why this happen.
Counter examples and also reference are welcome.
Thank You. 
 A: An outer measure is a set function which is not necessarily a measure as it is monotonic and subadditive, but not necessarily additive. Outer measures are usually defined an all subsets of the underlying set. For example, the Lebesgue outer measure is defined on all subsets of the set of reals. However, as I mentioned, it is only a subadditive function of sets, so for two arbitrary non-overlapping sets $A,B\subseteq \Bbb R$ you can only say that $\lambda^*(A\cup B) \leq \lambda^*(A) + \lambda^*(B)$ (subadditivity), whereas if $A,B$ are measurable, you have a stronger and more natural additivity property: $\lambda(A\cup B) = \lambda(A) + \lambda(B)$. Here $\lambda$ is the Lebesgue measure (say on Borel sets) and $\lambda^*$ is its outer measure on all subsets of reals.
A: Note that the Lebesgue outer measure is usually constructed as follows:

Let $m : \mathcal{R}(\mathbb{R}^n) \to \overline{\mathbb{R}}$ where $\mathcal{R}(\mathbb{R}^n)$ is the set of all $n$ dimensional rectangles so that
  $$
[a_1, b_1] \times [a_2, b_2] \times \cdots \times [a_n, b_n] \mapsto \prod_{i=1}^n (b_i - a_i)
$$
  then we put the Lebesgue outer measure $\mu^* : 2^{\mathbb{R}^n} \to \overline{\mathbb{R}}$ defined as
  $$
\mu^*(A) = \inf \left\{ \sum_i m(B_i) \biggm| A \subset \bigcup_i B_i , \{ B_i \} \subset \mathcal{R}(\mathbb{R}^n) \right\}
$$

So as you can see the outer measure is defined on all subsets of $\mathbb{R}^n$, but note that we then construct the sigma algebra $\mathcal{L}(\mathbb{R}^n)$ as

$
A \in \mathcal{L}(\mathbb{R}^n) \iff \forall S \subset \mathbb{R}^n$
  $$
\mu^*(S) = \mu^*(S \setminus A) + \mu^*(S \cap A)
$$

and then we can construct a measure space from the measurable space $(\mathbb{R}^n, \mathcal{L}(\mathbb{R}^n))$ as defining 

$\mu : \mathcal{L}(\mathbb{R}^n) \to \overline{\mathbb{R}}$ such that
  $$
A \in \mathcal{L}(\mathbb{R}^n) \mapsto \mu^*(A)
$$

Which I believe is what you're actually referring to as the outer measure on $\mathbb{R}$, since indeed there exist sets in $2^{\mathbb{R}^n}$ which aren't in $\mathcal{L}(\mathbb{R}^n)$, for example the Vitali Set.
