characteristic function/ Riemann-integrable

Question

Given $$A=\bigcup_i^n [a_i,b_i) \subset \mathbb{R} $$

Why is $$ \chi_A(x) $$ riemann integrable?

Answer 1

Riemman integrability makes sence in bounded intervals.

So if this union is contained in an interval $[a,b]$ and since the intervals in your union are disjoint and bounded,we have that the function $X_A=\sum_{i=1}^nX_{[a_i,b_i)}$ is piecewsise continuous and thus Riemman integrable and $$\int_a^bX_A(x)dx=\sum_{i=1}^n \int_{a_i}^{b_i}dx=\sum_{i=1}^n (b_i-a_i)$$

Answer 2

You can prove by induction on $n$ that $A$ can be written

$$A=\bigcup_j^m [a_j^\prime,b_j^\prime)$$ where

$\{a_1^\prime, \dots, a_m^\prime\} \subseteq \{a_1, \dots, a_n\}$ and $\{b_1^\prime, \dots, b_m^\prime\} \subseteq \{b_1, \dots, b_n\}$ and the interval $[a_j^\prime, b_j^\prime)$ are disjoint. So without loss of generality, we can suppose that the $[a_i,b_i)$ are disjoint.

Now consider the partition $\mathcal P \equiv a_1 < b_i < a_2 < b_2 < \dots <a_n < b_n$ and denote $a=a_1$, $b=b_n$ and $s = \sum_{i=1}^n (b_i - a_i)$. For any refinement $\mathcal P^\prime \equiv c_1 < \dots <c_k$ of $\mathcal P$ and $r_j \in (c_{j+1},c_j)$ we have $$\displaystyle \sum_{j=1}^{k-1} \chi_A(r_j)(c_{j+1}-c_j) = s$$

proving that $\chi_A$ is Riemann integrable.