PhyslibAlpha.Mathematics.MeasureTheory.LiftingAndConditioning

Let $X$ be a compact topological space and $Y$ be a topological space. Let $q : X \to Y$ be a continuous map and $\mu$ be a probability measure on $Y$. If the complement of the range of $q$ has $\mu$-measure zero (i.e., $\mu((\text{range } q)^c) = 0$), then there exists a probability measure $\nu$ on $X$ such that the pushforward measure of $\nu$ under $q$ is equal to $\mu$ (i.e., $q_* \nu = \mu$).

Given a probability measure $\rho$ on a space $A$ and two $\rho$-almost everywhere measurable maps $f, g : A \to B$, the average map is a probability measure on $B$ defined by the arithmetic mean of the pushforward measures $f_*\rho$ and $g_*\rho$: $$\frac{1}{2} f_*\rho + \frac{1}{2} g_*\rho$$ where $f_*\rho$ (respectively $g_*\rho$) denotes the image measure of $\rho$ under the map $f$ (respectively $g$).

Let $\rho$ be a probability measure on a space $A$, and let $f, g : A \to B$ be $\rho$-almost everywhere measurable maps. Let $u : B \to \mathbb{R}$ be a function such that $u$ is integrable with respect to the pushforward measures $f_*\rho$ and $g_*\rho$. Then the integral of $u$ with respect to the average measure $\frac{1}{2} f_*\rho + \frac{1}{2} g_*\rho$ is given by: $$\int_B u(x) \, d\left(\frac{1}{2} f_*\rho + \frac{1}{2} g_*\rho\right)(x) = \frac{1}{2} \int_A u(f(x)) \, d\rho(x) + \frac{1}{2} \int_A u(g(x)) \, d\rho(x)$$

Given a probability measure $\mu$ on a space $B$ and a set $S \subseteq B$ with non-zero measure (i.e., $\mu(S) \neq 0$), this definition constructs the conditional probability measure $\mu(\cdot | S)$, which represents the conditional law of $\mu$ restricted to the event $S$.

Let $\mu$ be a probability measure on a measurable space $B$, and let $S \subseteq B$ be a measurable set such that $\mu(S) \neq 0$. Then the conditional probability measure $\mu(\cdot | S)$ assigns zero measure to the complement of $S$, i.e., $\mu(S^c | S) = 0$.

Let $\mu$ be a probability measure on a measurable space $B$. For any measurable set $S \subseteq B$ with non-zero measure (i.e., $\mu(S) \neq 0$), the product of the scalar $\mu(S)$ and the conditional probability measure $\mu(\cdot | S)$ is equal to the restriction of the measure $\mu$ to the set $S$, denoted $\mu|_S$. In other words, $\mu(S) \cdot \mu(\cdot | S) = \mu|_S$.

Given probability measures $\mu$ and $\eta$ on a measurable space $B$ and a measurable set $S \subseteq B$, this definition constructs a new probability measure by replacing the mass of $\mu$ on $S$ with the measure $\eta$ scaled by the total mass $\mu(S)$. The resulting measure is defined as: $$\mu|_{S^c} + \mu(S) \eta$$ where $\mu|_{S^c}$ denotes the restriction of $\mu$ to the complement of $S$. This operation ensures that the total mass remains $1$, as the original mass $\mu(S)$ is redistributed according to the probability distribution $\eta$.

Let $\mu$ and $\eta$ be probability measures on a measurable space $B$, and let $S \subseteq B$ be a measurable set. The measure corresponding to the probability measure constructed by replacing the mass of $\mu$ on $S$ with $\eta$ is equal to the restriction of $\mu$ to the complement of $S$ (denoted $S^c$) plus the measure $\eta$ scaled by the mass $\mu(S)$. In mathematical notation: $$\text{replacePart}(\mu, \eta, S) = \mu|_{S^c} + \mu(S) \eta$$ where $\mu|_{S^c}$ is the restriction of the measure $\mu$ to the set $B \setminus S$.

Let $\mu$ be a probability measure on a measurable space $B$, and let $S \subseteq B$ be a measurable set such that $\mu(S) \neq 0$. For any function $f: B \to \mathbb{R}$ that is integrable with respect to $\mu$, the integral of $f$ over $B$ can be decomposed as the sum of the integral over the complement $S^c$ and the mass of $S$ multiplied by the conditional integral over $S$: $$\int_B f \, d\mu = \int_{S^c} f \, d\mu + \mu(S) \int_B f \, d\mu(\cdot | S)$$ where $\mu(\cdot | S)$ denotes the conditional probability measure given $S$.

Let $B$ be a measurable space. Let $\mu$ and $\eta$ be probability measures on $B$, and let $S \subseteq B$ be a measurable set. Suppose $f: B \to \mathbb{R}$ is a function that is integrable with respect to both $\mu$ and $\eta$. If we define the probability measure $\nu = \text{replacePart}(\mu, \eta, S)$ by replacing the mass of $\mu$ on $S$ with the measure $\eta$ scaled by $\mu(S)$ (i.e., $\nu = \mu|_{S^c} + \mu(S) \eta$), then the integral of $f$ with respect to $\nu$ is given by: $$\int_B f \, d\nu = \int_{S^c} f \, d\mu + \mu(S) \int_B f \, d\eta$$ where $\mu|_{S^c}$ denotes the restriction of $\mu$ to the complement of $S$.