Let $f\in C[a,b]$, 若 $\alpha(x)$ 是 increasing step function with finitely many discontinuous points
$x_1<x_2<\ldots<x_n$ in $(a,b)$, 則
$\int_a^b f(x) d\alpha(x)\\=f(a)(\alpha(a^+)-\alpha(a)) + \sum_{j=1}^nf(x_j)(\alpha(x_j^+)-\alpha(x_j^-)) + f(b)(\alpha(b)-\alpha(b^-))$
設 $\alpha(x)\in BV[a,b]$, $f(x)\in C[a,b]$, 如何定義 $\int_a^b f(x)d\alpha(x)$ ?
由於 Functions of Bounded Variation as a Difference of Two Increasing Functions
我們可以將 $\alpha$ 拆解成兩個 increasing function 差. 注意其分解法不唯一, 任取 $\beta_1, \beta_2 \uparrow$ such that $\alpha(x)=\beta_1(x)-\beta_2(x)$
且由於 Integrand 連續 Integrator 遞增, 則 RS Integral on [a, b], $\int_a^b f(x) d\beta_1(x)$ and $\int_a^b f(x) d\beta_2(x)$ 存在, 則我們可令
$\int_a^b f(x) d\alpha(x)=\int_a^b f(x) d\beta_1(x)-\int_a^b f(x) d\beta_2(x) \ldots (\star)$
證明定義 $(\star)$ 與 $\alpha$ 的分解方式無關
求 $\int_0^n x^2 d[x^2]$ ? ($[x]$ means step function)
設 $f(x) \in C^1[a,b]$, 求 $V(f, [a,b])$