A measure-theoretic definition
Dr. Krishna Kumar
The Polaris Advisor Program
© 2025 Dr. Krishna Kumar, All rights reserved.
We assume that all reasoning takes place over the real line \(\mathbb{R}\) equipped with the Borel \(\sigma\)-algebra \(\mathcal{B}\) and the Lebesgue measure \(\lambda\). This gives us a standard measurable space \((\mathbb{R}, \mathcal{B}, \lambda)\).
All presence reasoning in the Presence Calculus is grounded in this measurable structure. When working with unbounded intervals or domains that may extend to \(\pm\infty\), we consider the extended real line \(\overline{\mathbb{R}} = \mathbb{R} \cup \{ -\infty, +\infty \}\). While \(\lambda\) is not defined on all subsets of \(\overline{\mathbb{R}}\), integration on \([a, +\infty)\) or \((-\infty, b]\) is well-defined for functions that decay appropriately.
Let
\[ F : (e, b, t) \to \mathbb{R} \]
be a function that assigns a value in \(\mathbb{R}\) to each element–boundary–time triple. This is the general form of a presence density function also called signal, in the Presence Calculus.
To define a presence over such a function, we require that \(F\) induces a measure over \(\mathbb{R}\): that is, for any fixed pair \((e, b)\), the function \(t \mapsto F(e, b, t)\) must be measurable with respect to the Borel \(\sigma\)-algebra \(\mathcal{B}\), and Lebesgue integrable over all finite intervals.
This guarantees that for any interval \([t_0, t_1) \subset \mathbb{R}\), the integral
\[ \mu(e, b, [t_0, t_1)) = \int_{t_0}^{t_1} \lvert F(e, b, t) \rvert \, dt \]
is well-defined and finite.
We do not require that \(F\) be jointly measurable over \((e, b, t)\), nor that \(E\) and \(B\) carry their own \(\sigma\)-algebras. It is sufficient that \(F(e, b, \cdot)\) be measurable for each fixed \((e, b)\).
Observation intervals are taken to be half-open intervals of the form \([t_0, t_1) \subset \mathbb{R}\). The collection of all such intervals forms a basis:
\[ \mathcal{B}_T = \{ [a, b) \mid a < b \in \mathbb{R} \} \]
This basis generates a topology \(\tau\) on \(\mathbb{R}\) via:
\[ \tau = \left\{ \bigcup \mathcal{F} \mid \mathcal{F} \subset \mathcal{B}_T \right\} \]
This topology is equivalent to the standard topology on \(\mathbb{R}\), since open intervals \((a, b)\) and half-open intervals \([a, b)\) generate the same collection of open sets under union.
This topological structure supports the finite additivity of presence mass by ensuring that unions, intersections, and partitions of presence intervals correspond to open sets in this topology, allowing the induced measures \(\mu_{e,b}\) to be consistently defined across composed regions of time.
For each fixed \((e, b)\), the function \(t \mapsto F(e, b, t)\) induces a measure on \(\mathbb{R}\) via:
\[ \mu_{e,b}(A) = \int_A F(e, b, t)\, dt \quad \text{for all } A \in \mathcal{B} \]
This measure \(\mu_{e,b}\) is finitely additive over measurable sets: if \(A\) and \(B\) are disjoint and measurable,
\[ \mu_{e,b}(A \cup B) = \mu_{e,b}(A) + \mu_{e,b}(B) \]
This follows from standard properties of the Lebesgue integral.
This ensures the presence accumulation recurrence is valid over entries in the presence accumulation matrix.