Definition
Let X be a separable, reflexive Banach space with norm || || and fix T > 0. Let BV−(X) denote the space of all left-continuous functions z : → X with bounded variation on .
For any function of time f, use subscripts +/− to denote the right/left continuous versions of f, i.e.
For any sub-interval of, let Var(z, ) denote the variation of z over, i.e., the supremum
The first step in the construction of the reduced derivative is the “stretch” time so that z can be linearly interpolated at its jump points. To this end, define
The “stretched time” function τ̂ is left-continuous (i.e. τ̂ = τ̂−); moreover, τ̂− and τ̂+ are strictly increasing and agree except at the (at most countable) jump points of z. Setting T̂ = τ̂(T), this “stretch” can be inverted by
Using this, the stretched version of z is defined by
where θ ∈ and
The effect of this definition is to create a new function ẑ which “stretches out” the jumps of z by linear interpolation. A quick calculation shows that ẑ is not just continuous, but also lies in a Sobolev space:
The derivative of ẑ(τ) with respect to τ is defined almost everywhere with respect to Lebesgue measure. The reduced derivative of z is the pull-back of this derivative by the stretching function τ̂ : → . In other words,
Associated with this pull-back of the derivative is the pull-back of Lebesgue measure on, which defines the differential measure μz:
Read more about this topic: Reduced Derivative
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