A partition of unity can be thought of as a blurred out set partition: identify the fibers of the set partition with their indicator functions; the sum of these indicators is $1$. I would like to extend the definition of entropy from set partitions to partitions of unity. My purpose is to capture the notion of information obtained by making noisy or imprecise measurements (I view ordinary set partitions as measurements that are quantized but noiseless). Also, this setup is a satisfying formulation of the idea of a *coarse-graining* in statistical physics.

Before giving the definitions, here is my question: **Do these generalized partitions give rise to equivalent results and definitions in the theory of dynamical systems?** For example, is it possible to modify the definition of the dynamical (Kolmogorov–Sinai) entropy to allow partitions of unity, and is the resulting definition equivalent? What about the Shannon–McMillan–Breiman Theorem? More broadly, **do they have any interesting new properties at all**? A negative answer would be interesting too, because it would mean that we aren't losing anything by working only with deterministic measurements. In addition, I'd be very interested to hear of any **existing work or connections to other areas**.

**Definitions:**

Given a probability space $(X, \mu)$, a (finite, labeled) **set partition** of $X$ is a finite indexed family of nonempty pairwise disjoint measurable sets whose union is $X$. Given a partition $\Pi$, the **information function** is the random variable $I_\Pi : X \to \mathbb R, \;\; x \mapsto -\log \mu \,E_x$ where $E_x$ is the unique element of $\Pi$ such that $x \in E_x$. The **static** (or **Shannon**) **entropy** of a set partition $\Pi$ is the expectation of the information function, $H(\Pi) := \mathbb E(I_\Pi) = -\sum_{E \in \Pi} \mu E \; \log \mu E$.

More generally, we can define a (measure-theoretic, finite) **partition of unity** on $X$ to be an indexed family of measurable functions $f_1, \ldots, f_n : X \to [0, 1]$, with $\sum_i f_i(x) = 1$ for every $x \in X$. Given a function $f : X \to \mathbb R_{\ge 0}$ whose integral is strictly positive, let
$$I(f) := \int \hat f(x) \, \log \hat f(x) \, \mathrm{d}\mu(x) \quad \text{where}\; \hat f = \frac{f} {\int f\,\mathrm{d}\mu}.$$
(There's no minus sign: not a typo. Heuristically speaking, $I(f)$ is the information obtained about $x$ by observing the event "$f$ occured"; cf. the definition of differential entropy.)

A partition of unity is essentially the same thing as a probability transition kernel $K$ from $X$ into $\{1, \ldots, n\}$: The measure $K(x, \cdot)$ on $\{1, \ldots n\}$ is the aforementioned noisy measurement of $x \in X$. Let $\Pi$ be a partition of unity and $K$ the associated kernel. Let the **static entropy** of $\Pi$ be the expectation of the pullback $K I$. Equivalently,
$$H(\Pi) \;:=\; \sum_{f \in \Pi} \left( \int f \mathrm{d}\mu \right) I(f).$$

Here are a few trivial properties, which the definitions above were meant to capture:

- The two definitions of $H$ are compatible: If $\Pi$ is the family of indicator functions of the fibers of a set partition, we recover the static entropy of the partition. This is because $I(\mathbf 1_E) = -\log \mu E$ for every measurable set $E \subseteq X$.
- $I(kf) = I(f)$ for every $k > 0$ and $f : X \to \mathbb R_{\ge 0}$ with $\int f \mathrm{d}\mu > 0$.
- $I(f) = 0$ whenever $f \equiv c > 0$. (A flat posterior density contains no information.)
- If $X = \mathbb R^d$ and $A$ is an invertible linear operator, then $I(f \circ A) = I(f) + \log |\det A|$ for every $f : X \to \mathbb R_{\ge 0}$ with $\int f \mathrm{d}\mu > 0$.

A few more definitions will need to be generalized, though I haven't worked out the rest yet. For example, one candidate for the join of two partitions of unity $\Pi_1, \Pi_2$ is the family of pointwise products, $(f_1 f_2)_{f_1 \in \Pi_1, f_2 \in \Pi_2}$ (this is a partition of unity because $\sum_{i,j} (f_{1i} f_{2j})(x)$ $=$ $\sum_i f_{1i}(x) \sum_j f_{2j}(x)$ $= 1$).