AbsolutelyContinuousMeasure

Absolutely Continuous Measure

In measure theory, a specialized area of mathematics that deals with sets, functions, and integrals, the concept of "absolutely continuous measure" plays a significant role. This concept formalizes the mathematical relationship between two measures in a way that one measure is "controlled" by another.

Let $\mu$ and $\nu$ be two measures defined on a measurable space $(X, \mathcal{F})$, where $X$ is the set and $\mathcal{F}$ is a σ-algebra on $X$.

A measure $\mu$ is said to be absolutely continuous with respect to another measure $\nu$, denoted as $\mu \ll \nu$, if for every measurable set $A \in \mathcal{F}$ such that $\nu(A) = 0$, it follows that $\mu(A) = 0$. In this formal relationship, $\mu$ is "controlled" by $\nu$ in the sense that $\mu$ assigns zero measure to all sets to which $\nu$ assigns zero measure.

Properties and Implications

1. Radon-Nikodym Theorem: A cornerstone in the study of absolutely continuous measures is the Radon-Nikodym theorem. This theorem states that if $\mu$ is absolutely continuous with respect to $\nu$, then there exists a measurable function $f: X \to [0, \infty)$ such that for every measurable set $A$

$$\mu(A) = \int_A f d\nu$$

2. Comparison to Other Types of Dominance: Absolute continuity is a stronger condition than other types of relationships between measures, such as "mutual singularity."

3. Change of Variables in Integration: The concept of absolute continuity extends to calculus, where it generalizes the idea of a change of variables in integration.

4. Applications: The notion of absolutely continuous measures is pivotal in various mathematical disciplines and applications, including but not limited to probability theory, statistics, functional analysis, and ergodic theory.

5. Transitivity: The property of absolute continuity is transitive. That is, if $\mu \ll \nu$ and $\nu \ll \lambda$, then $\mu \ll \lambda$.

6. Uniqueness: If $\mu \ll \nu$ and $\mu$ is a finite measure, then the Radon-Nikodym derivative $f$ is unique up to a $\nu$-null set.

Theorems Involving Absolute Continuity in Measure Theory and Stochastic Calculus

Radon-Nikodym Theorem

Suppose $\nu$ and $\mu$ are $\sigma$-finite measures on a measurable space $(X, \mathcal{F})$, and $\nu$ is absolutely continuous with respect to $\mu$ (written $\nu \ll \mu$). Then there exists a unique $\mu$-integrable function $f: X \to [0, \infty)$ such that for every measurable set $A \in \mathcal{F}$,

$$\nu(A) = \int_A f d\mu.$$

The function $f$ is called the Radon-Nikodym derivative of $\nu$ with respect to $\mu$, denoted $\frac{d\nu}{d\mu}$.

Cameron-Martin Theorem

This theorem characterizes the set of absolutely continuous changes of measure on Wiener space. It provides the Radon-Nikodym derivative $\frac{d\mu_h}{d\mu}$ for a translated measure $\mu_h$ in terms of the original Wiener measure $\mu$.

Girsanov's Theorem

Suppose $W_t$ is a standard Brownian motion under a probability measure $P$, and let $X_t$ be a stochastic process adapted to the same filtration as $W_t$ such that $\int_0^T X_t^2 dt < \infty$ almost surely for all $T > 0$. Define a new measure $Q$ by the Radon-Nikodym derivative

$$\frac{dQ}{dP} = \exp\left( -\int_0^T X_t dW_t - \frac{1}{2} \int_0^T X_t^2 dt \right).$$

Then under $Q$, the process $W_t - \int_0^t X_s ds$ is a standard Brownian motion.

Tying Them Together

1. Radon-Nikodym Theorem: Provides the mathematical foundation for talking about the derivative of one measure with respect to another. This is the basis for both the Cameron-Martin and Girsanov theorems, which give specific forms of the Radon-Nikodym derivative in the context of stochastic processes.

2. Cameron-Martin Theorem: Applies the Radon-Nikodym concept to Wiener space, providing the Radon-Nikodym derivative for a translated measure $\mu_h$ in terms of the original Wiener measure $\mu$.

3. Girsanov's Theorem: Extends the idea further by allowing for a change of measure that turns one Brownian motion into another, under a drift term $X_t$. The Radon-Nikodym derivative $\frac{dQ}{dP}$ is explicitly given, allowing us to work under the new measure $Q$.

In summary, the Radon-Nikodym Theorem provides the general framework for changes of measure, the Cameron-Martin Theorem applies this to Wiener space and Brownian motion, and Girsanov's Theorem extends this to more general changes of measure involving Brownian motion with drift.