5.5 Riemann-Stieljes Integral
Let \(f,\, \alpha \) be functions defined on the interval \([a,b]\). We shall assume through out the discussion that \(\alpha \) is an
increasing function and \(f\) is bounded. Let \(P:\, a = x_0 < x_1 < \cdots \cdots \cdots < x_n = b\) be a partition of \([a,b]\). We define \(\, M_k(f) = \sup \big \{f(x):\, x_{k -1} < x < x_k\big \}\hspace {0.3cm},\\ \hspace {0.3cm} m_k = \inf \big \{f(x):\, x_{k-1} < x < x_k\big \}\). Note that since \(f\) is bounded, \(M_k(f)\)
and \(m_k(f)\) are finite. Let \(t_k\) be a point in \([x_{k-1},x_k]\hspace {0.2cm} k = 1,\, 2,\, \cdots \cdots ,\, n\).
Define \(\Delta \alpha _k = \alpha (x_k) - \alpha (x_{k - 1})\hspace {0.2cm},\hspace {0.2cm} k = 1,\, 2,\, \cdots \cdots , n\)
The upper sum associated with the partition \(P, \, f,\, \alpha \,\) is defined as \(\,U(P, f, \alpha ) = \displaystyle {\sum ^n_{k =1}\, M_k(f)\, \Delta \alpha _k}\)
Similarly, the lower sum is \(\hspace {0.2cm} \displaystyle {L(P, f, \alpha ) = \sum ^n_{k = 1}\, m_k(f)\, \Delta \alpha _k}\).
The Riemann-Stieljes sum is \(\hspace {0.2cm} \displaystyle {S(P, f, \alpha ) = \sum ^n_{k = 1}\, f(t_,)\, \Delta \alpha _k}\)
Definition 5.5.1. If \(f:\, [a,b]\longrightarrow \mathbb {R}\) is bounded function such that \(U(P, f, \alpha )\) and \(L(P, f, \alpha )\) approach a common finite limit as \(\begin {vmatrix} \begin {vmatrix} P\\ \end {vmatrix} \end {vmatrix}\longrightarrow 0\),
then we say that \(f\) is Riemann-Stieljes integrable with respect to \(\alpha \) or that the Riemann-Stieljes
integral of \(f\) with respect to \(\alpha \) exists. The integral is the denoted by \(\hspace {0.2cm} \displaystyle {\int ^b_af(x)\, d\alpha (x)\hspace {0.3cm}\text {or}\hspace {0.3cm} \int ^b_af\, d\alpha }\).
If \(\alpha (x) = \alpha \) then \(\hspace {0.2cm}\displaystyle {\int ^b_af\,d\alpha }\hspace {0.2cm}\) is the used Riemann integral of \(f\) denoted by \(\hspace {0.2cm}\displaystyle {\int ^b_af(x)\, dx}\).
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