A measure-theoretic generalization
Dr. Krishna Kumar
The Polaris Advisor Program
© 2025 Krishna Kumar. All rights reserved.
When generalizing from a single presence density function to a system of presences, it is useful to think of presence as a quantity distributed over a product space formed from two orthogonal domains:
We define the product measurable space: \[ (T \times S, \mathcal{F}_T \otimes \mathcal{F}_S) \] and the product measure: \[ \mu = \mu_T \otimes \mu_S. \]
Let \(f: T \times S \to \mathbb{R}_{\geq 0}\) be a measurable function representing presence density over this joint domain.
The total presence mass accumulated across the full product space is defined as: \[ A = \\\iint_{T \times S} f(t, s)\\, d\mu_T(t)\\, d\mu_S(s). \]
This integral assumes that \(f\) is integrable under the product measure \(\mu\), i.e.: \[ \\\iint |f(t, s)| \\, d\mu_T(t)\\, d\mu_S(s) < \\\infty. \]
We define marginal densities over each domain:
Presence per unit time: \[ D_T(t) = \\\int_S f(t, s)\\, d\mu_S(s). \]
Presence per signal: \[ D_S(s) = \\\int_T f(t, s)\\, d\mu_T(t). \]
Intuitively, marginalization distributes total presence mass along the time or signal dimension by integrating out the other variable.
From a statistical lens, we are accustomed to thinking of these quantities as averages. Time averages, however, often rely on assumptions like ergodicity or stationarity of some underlying distribution over which the averages are computed.
The measure-theoretic definition via marginal densities avoids these assumptions and provides a structurally invariant description of presence across domains. Rather than thinking of both quantities as averages, it is more general to treat them as densities.
Note: The averages \(\bar{D}_T\) and \(\bar{D}_S\) are only well-defined when \(\mu_T(T)\) and \(\mu_S(S)\) are finite. This ensures that the total presence mass can be meaningfully distributed over each domain.
In the common case where the signal domain is finite or countable (e.g., a discrete set of labeled signals) and the time domain is also quantized (e.g., measured in ticks or time buckets as in the presence matrix), \(\bar{D}_S\) corresponds to the average presence per signal, while \(\bar{D}_T\) captures the average presence per unit time.
These quantities let us view the system from either a signal-centric or time-centric perspective, without changing the total mass. In either case, the mass is obtained by integrating over a presence density function.
Define the average presence densities: \[ \\\bar{D}_T = \\\frac{A}{\mu_T(T)}, \quad \\\bar{D}_S = \\\frac{A}{\mu_S(S)}. \]
Then the generalized presence invariant becomes: \[ \\\frac{\\\bar{D}_S}{\\\bar{D}_T} = \\\frac{\mu_T(T)}{\mu_S(S)}. \]
This expresses that the average presence densities in the time and signal domains are proportionally related, and that the constant of proportionality is the ratio of the domain measures.
In many practical settings, particularly when working with discrete signal domains, the signal measure \(\mu_S\) is a counting measure. In such cases, if \(N = \mu_S(S_T)\) and \(T = \mu_T(T)\), then the ratio: \[ \\\frac{N}{T} = \\\frac{\mu_S(S_T)}{\mu_T(T)} \] represents the ratio of the “sizes” of the sets over which presence mass is being distributed. That is:
Presence mass exists jointly over time and signal. When we marginalize that mass, we can average it over the extent of time or the extent of signals. The ratio ( N/T ) tells us how these marginal densities relate — it quantifies the structural “rate” or “density of signals per unit time” induced by the co-presence.
This view aligns with the intuition of the Radon–Nikodym derivative, where presence densities reflect the relative rate of one measure with respect to another across their domain of interaction.
If \(f: T \times S \to \mathbb{R}\) is integrable over the product measure \(\mu_T \otimes \mu_S\), then: \[ \\\iint_{T \times S} f(t, s) \\, d\mu_T(t)\\, d\mu_S(s) = \\\int_T \left( \\\int_S f(t, s) \\, d\mu_S(s) \right) d\mu_T(t) = \\\int_S \left( \\\int_T f(t, s) \\, d\mu_T(t) \right) d\mu_S(s). \]
This justifies that presence mass can be marginalized in either order, and confirms the internal consistency of the invariant.
The invariant tells us:
Presence mass is conserved, and marginal densities are scaled by the ratio of the measures of their support domains. This allows us to move coherently between signal-normalized and time-normalized views of system behavior.