Regulated integral

Definition

Definition on step functions

Extension to regulated functions

• there is a sequence of step functions _n∈N such that || φ_n − f ||_∞ → 0 as n → ∞; or, equivalently,
• for all ε > 0, there exists a step function φ_ε such that || φ_ε − f ||_∞ < ε; or, equivalently,
• f lies in the closure of the space of step functions, where the closure is taken in the space of all bounded functions → R and with respect to the supremum norm || - ||_∞; or equivalently,
• for every t ∈ a, b), the [right-sided limit

Properties of the regulated integral

• The integral is a linear operator: for any regulated functions f and g and constants α and β,
• The integral is also a bounded operator: every regulated function f is bounded, and if m ≤ f ≤ M for all t ∈ , then
• Since step functions are integrable and the integrability and the value of a Riemann integral are compatible with uniform limits, the regulated integral is a special case of the Riemann integral.

  Extension to functions defined on the whole real line

• the partition on whose open intervals a step function is required to be constant is allowed to be a countable set, but must be a discrete set, i.e. have no limit points;
• the requirement of uniform convergence must be loosened to the requirement of uniform convergence on compact sets, i.e. closed and bounded intervals;
• not every bounded function is integrable. This leads to a notion of local integrability.

  Extension to vector-valued functions