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A regulated function is defined as a function from a real interval $I \subset \mathbb R$ to $E$, a complete real or complex normed (Banach) space which satisfies certain conditions as follows.

Some authors restrict I to be a closed bounded interval: Wiki; Bartle – modern theory of integration p.48; Banas & Kot – On Regulated Functions.27; Ingleton – Notes on integration.
Others allow any interval including $(-\infty, \infty)$: Nicolas Bourbaki - Functions of a Real Variable p. 53; Dieudonné - Foundations of modern analysis p. 145. (note that Dieudonné was a member of the Bourbaki collective).

When a closed bounded interval is chosen then regulated functions can be equivalently defined as the completion of the space of step functions with the supremum norm, or as functions which have a left and right limit at every point in the interval (with obvious amendment for the endpoints). Clearly a regulated function on a bounded interval is bounded as there must be a (necessarily bounded) step function close to it.

The two definitions are not equivalent for an unbounded interval, for example, the function $f: (0, \infty) \to \mathbb R$ defined as $f(x) = x$. This is continuous at every point of $(0, \infty)$ but is there is no uniformly convergent sequence of step functions for it (Proof: suppose there were. Then there is a necessarily bounded step function close to it $\implies f(x)$ bounded – contradiction). When an unbounded interval is allowed Dieudonné sticks with the left/right limit definition and shows equivalence of uniform convergence of step functions in the case that the interval is bounded. Bourbaki on the other hand refer to a space “$\mathscr F (I; E)$ of maps from $I$ into $E$, endowed with the topology of uniform convergence on every compact subset of I (Gen. Top., X, p. 277)”. They then define a function as regulated if it is the uniform limit of step functions on every compact subset of $I$ (which one sees includes the continuous function $f(x) = x$ in the example above). And they then note that the space of regulated functions is the closure of the space of step functions as a subspace of $\mathscr F $. In this definition, a regulated function need not be bounded (on an unbounded interval).


Questions:

  1. Regulated functions include functions of bounded variation on a closed interval and are a subset of Riemann integrable functions on a closed interval, so is the more general definition on an unbounded interval of practical value ?
  2. I don't have access to Bourbaki's Gen. Top., X, p. 277 and would like to know what thier space $\mathscr F$ is about.
  3. Any other light on this topic would also be appreciated.
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