# リーマン積分

## Definition リーマン積分

Suppose $$I = [a_1, b_1] \times [a_2, b_2] \times \cdots \times [a_d, b_d] \in \mathbb{R}^d$$ then $|I|$ (measure of $I$) defined below $$|I| = (a_1, b_1) (a_2, b_2) \cdots (a_d, b_d)$$ * 各区間が開区間、半閉半開区間であっても $|I|$ は同じく定義される.
Suppose $$f:\mathbb{R}^d \rightarrow \mathbb{R}$$ and

$$\overline{f_j} = \sup_{x \in I_j} f(x) \\ \underline{f_j} = \inf_{x \in I_j} f(x) \\ \overline{s}(f,\Delta) = \sum_{j=1}^k \overline{f_j} |I_J| \\ \underline{s}(f,\Delta) = \sum_{j=1}^k \underline{f_j} |I_J| \\$$ と定義する.
Darboux(ダルブー)の上積分、下積分 is defined by $$\overline{\int}_I f dx = \inf_{\Delta} \overline{s}(f,\Delta) \\ \underline{\int}_I f dx = \sup_{\Delta} \underline{s}(f,\Delta) \\$$ If $$\overline{\int}_I f dx = \underline{\int}_I f dx\\$$ then
Riemann積分 is defined by $$\mathcal{R} \int_I f dx = \overline{\int}_I f dx = \underline{\int}_I f dx\\$$