Clarification of proof involving characteristic function in $L^1$

Question

Can someone help me please to clarify how is this true in $L^1$.

when calculating the integral I can't see how the result gives an $h$. Thanks in advance.

Answer 1

If $I$ has end points $a$ and $b$ then $\int|\chi_I(x+h)-\chi_I(x)|dx=\int_{a-h}^a dx+\int_b^{b+h}dx=|h|+|h|=2|h|$ provided $h >0$ and $h<b-a$. For $a-b <h<0$ a similar argument works. For $|h| >b-a$ this is false.