Integral of the characteristic function.

Question

I know $\chi$ is the characteristic function.

Why does $\int \chi_I= L(I)$? where L(I) is the length of I.

Answer 1

The definition of the Lebesgue integral with measure $\mu$ is that

$$\int \chi_I d\mu = \mu(I)$$

Assuming that your measure is just the usual Lebesgue measure on $\mathbb{R}$, and $I$ is an interval, then $\mu(I)$ is its length. If $I$ is a more general set, then its length is defined to be its Lebesgue measure.