Example of a subset of $\mathbb{R}$ which is not measurable? Can anyone please give me an example of non-measurable real set? 
 A: The standard example is the Vitali Set. It is constructed from the equivalence relation $x\sim y \leftrightarrow x-y\in\mathbb{Q}$, where $\mathbb{Q}$ is the set of rationals.
Consider a set $V$ which is a complete set of representatives of these equivalence classes all in the interval $[0,1]$. That is, pick one element from each equivalence class (this requires the Axiom of Choice) and consider it mod 1. The restriction of the set to this interval does not pose any problems because adding or subtracting 1 to an element of $\mathbb{R}$ keeps it in the same equivalence class. 
To show that $V$ is non-measurable, note that all translations of $V$ mod 1 by a rational number $q \in [0,1)$ are disjoint (because the addition of $q$ will keep each element in its own equivalence class), and the union of all such translations is $[0,1]$. Suppose $V$ has Lebesgue measure $m(V)$. Then by the countable additivity of the Lebesgue measure,
$$
m([0,1]) = m\left(\bigcup_{q\in\mathbb{Q}\cap[0,1)} V+q\right) = \sum_{k=1}^{\infty} m(V).
$$
The left-hand side is equal to 1, but the right-hand side must be either 0 or $\infty$. Thus $V$ is not measurable.
