INTEGRATION OF VECTOR VALUED FUNCTIONS

This lecture discusses the integration of vector-valued functions and extending the Riemann-Stieltjes integral beyond monotonically increasing functions $\alpha$.
A vector-valued function $f$ from $[a, b]$ to $\mathbb{R}^k$ gives rise to $k$ scalar-valued functions $f_1, f_2, \dots, f_k$ from $[a, b]$ to $\mathbb{R}$.
The integral of $f$ with respect to a monotonically increasing $\alpha$ is defined as a vector in $\mathbb{R}^k$ with components being the Riemann-Stieltjes integrals of $f_i$ with respect to $\alpha$:$$\int_a^b f d\alpha = \left( \int_a^b f_1 d\alpha, \int_a^b f_2 d\alpha, \dots, \int_a^b f_k d\alpha \right)$$
For a vector-valued function $f$, the property relating the norm of the integral to the integral of the norm is: $||\int_a^b f d\alpha|| \le \int_a^b ||f|| d\alpha$. If $f$ is integrable, the real-valued function $||f||$ is also integrable.
The class of BV functions extends the class of all monotonic functions, allowing for the integration of $f d\alpha$ when $\alpha$ is not monotonically increasing, as needed for concepts like integration by parts.
A function $\alpha$ from $[a, b]$ to $\mathbb{R}$ is of bounded variation if the supremum of $\sum_{i=1}^{n} |\alpha(x_i) - \alpha(x_{i-1})|$ over all partitions $P$ of $[a, b]$ is finite. This supremum is called the total variation.
All monotonic functions are of bounded variation. The total variation of a monotonic function $\alpha$ is $|\alpha(b) - \alpha(a)|$.
If $\alpha$ is differentiable on $[a, b]$ and its derivative $\alpha'$ is bounded, then $\alpha$ is a function of bounded variation.
The set of BV functions, denoted $BV([a, b])$, forms a vector space.
A function $\alpha$ of bounded variation can be represented as the difference of two monotonically increasing functions, $\alpha = \alpha_1 - \alpha_2$.
The integral $\int_a^b f d\alpha$ for $\alpha \in BV([a, b])$ is defined as the difference of two Riemann-Stieltjes integrals: $\int_a^b f d\alpha_1 - \int_a^b f d\alpha_2$.
For a curve $\gamma$ in $\mathbb{R}^k$, the total variation of $\gamma$ corresponds to the length of the curve.