Extension of the Lebesgue measure on the extended real line

Question

Is it possible to extend the Lebesgue measure on $\mathbb{R}$ as a measure on $[-\infty,\infty]$ (defined on the Borel subsets of $[-\infty,\infty]$), so that $[-\infty,\infty]\backslash \mathbb{R}$ is a null set ?

Answer 1

As proposed in the comment, we can just "extend by zero sets." That is, since we have only added two points it is intuitive to consider them as negligible. So, if $(\mathbb{R},\Omega, \lambda)$ is your initial measure space - where $\lambda$ denotes the Lebesgue measure, define $(\widehat{\mathbb{R}}, \widehat{\Omega}, \mu)$ by declaring a set $A$ to be measurable in $\widehat{\mathbb{R}}$ if and only if $A\cap \mathbb{R}$ is Lebesgue Measurable. Further, define $$ \mu(A)=\lambda(A\cap \mathbb{R}).$$ It's not so hard to check that this provides a measure. It solves your problem because $X=[-\infty,\infty]\setminus \mathbb{R}=\{\pm \infty\}$, and $$\mu(\{\pm\infty\})=\lambda(\{\pm\infty\cap \mathbb{R}\})=\lambda(\varnothing)=0.$$ You might also notice that this technique can be generalized.