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I know $\chi$ is the characteristic function.

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

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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.

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