Intuizione dietro la Misura di Lebesque

[que1]

Intuitivamente quale è il significato ( nel caso Reale) di misura di un insieme ? Lo chiedo perchè non
riesco a spiegarmi intuitivamente il criterio di integrabilità di Riemann secondo Lebesque.
Grazie

[javicemarpe]

The only thing you have to understand about Lebesgue measure in order to understand the Riemann integrability criteria is the meaning of zero measure sets, and this is clear. A subset of $\mathbb{R}$ is a zero measure set if you can cover it with intervals of measure as small as you want.

In general, the concept of Lebesgue measure of a set is a generalisation of the concept of longitude of an interval and the way we generalise this concept is by covering a set with intervals. We define the measure of a set as the less sum of the longitudes of intervals covering the set.

For example, a finite subset $A$ of $\mathbb{R}$ has finite Lebesgue measure. Indeed, let us write $A=\{x_1,\ldots,x_n\}$. Then we can cover this set with the the family of intervals $\mathcal{U}_\varepsilon=\{(x_i-\varepsilon/{2n},x_i+\varepsilon/{2n})\}_{i=1}^n$ for all $\varepsilon>0$, and the sum of the longitudes of the intervals of these families is $\varepsilon$. Then, you can cover $A$ with intervals of very small longitude and, then, $A$ is a zero measure set.

[que1]

thanks