# ルベーグ積分

## ルベーグ積分の定義

Suppose $$\lim_{p \rightarrow \infty} s_p(x) = f(x) \\ (\forall i,j \in \mathbb{N}, 0 \le s_i(x), i \lt j \Rightarrow s_i(x) \le s_j(x))$$ then $$\int_{R^d} f(x)dx := \lim_{p \rightarrow \infty}\int_{R^d} s_p(x) dm(x)$$ where $$\int_{R^d} s(x)dm(x) := \sum_{j=1}^k a_j m(E_j)$$ where $$s(x) = \sum_{j=1}^k a_j \chi_{E_j}(x)$$ where $$\forall i,j \in \{1,2,\dots k\}, E_j \in \mathcal{M}(R^d), a_j > 0, i \neq j \Rightarrow (E_i \cap E_j = \emptyset , a_i \neq a_j)$$ とする