The concept of equivalence of measures plays a crucial role in probability theory and measure theory, providing a framework to compare and relate different ways of assigning probabilities or measuring sets. Two measures are considered equivalent if they agree on the "size" of sets, even if they might assign different numerical values to these sets. This notion allows us to switch between different measures while preserving essential properties like probability calculations or integration. In this article, we will delve into the formal definition of equivalence of measures, explore its significance, and illustrate its application through examples.
Formal Definition
Equivalence of measures is defined in terms of absolute continuity. Let $(\Omega, \mathcal{F})$ be a measurable space, where $\Omega$ is the sample space and $\mathcal{F}$ is a sigma-algebra of events. Two measures, $\mu$ and $\nu$ defined on this measurable space are said to be equivalent if they satisfy the following conditions:
- Absolute Continuity: $\mu$ is absolutely continuous with respect to $\nu$ (denoted as $\mu \ll \nu$), meaning that if $\nu(A) = 0$ for some $A \in \mathcal{F}$, then $\mu(A) = 0$. In other words, if a set has zero measure under $\nu$, it also has zero measure under $\mu$.
- Mutual Absolute Continuity: $\nu$ is also absolutely continuous with respect to $\mu$ ($\nu \ll \mu$).
Therefore, two measures are equivalent if they agree on which sets have zero measure.
Significance of Equivalence of Measures
The concept of equivalence of measures holds significant importance in various areas of mathematics:
1. Probability Theory:
In probability theory, equivalence of measures allows us to change the probability measure without altering the probabilities of events. This is particularly useful when dealing with transformations of random variables or when we want to work with a more convenient probability measure for calculations.
2. Measure Theory:
In measure theory, equivalence of measures allows us to define integrals with respect to different measures. This is essential for studying properties of functions and sets in a more general setting.
3. Stochastic Processes:
Equivalence of measures is also crucial in the study of stochastic processes. It helps us understand how different probability measures can be used to describe the evolution of a system, particularly when dealing with changes in time or changes in parameters.
Examples of Equivalence of Measures
To illustrate the concept of equivalence of measures, consider the following examples:
1. Discrete Probability Distributions:
Let $\Omega = {1, 2, 3}$ be a sample space, and let $\mu$ be the uniform probability measure on $\Omega$, i.e., $\mu(1) = \mu(2) = \mu(3) = 1/3$. Now, consider another probability measure $\nu$ defined as follows:
$\nu(1) = 1/4$, $\nu(2) = 1/2$, and $\nu(3) = 1/4$.
Notice that $\mu$ and $\nu$ assign different probabilities to individual outcomes, but they agree on which sets have zero probability. For example, both $\mu(\emptyset) = 0$ and $\nu(\emptyset) = 0$. Therefore, $\mu$ and $\nu$ are equivalent measures.
2. Continuous Probability Distributions:
Consider the case of two continuous probability distributions on the real line, a standard normal distribution with probability density function $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ and an exponential distribution with density function $f(x) = \lambda e^{-\lambda x}$ for $x \ge 0$. These two distributions are not equivalent because the exponential distribution assigns positive probability to the set of negative real numbers, while the standard normal distribution assigns zero probability to this set. Therefore, they do not satisfy the condition of absolute continuity.
Conclusion
The concept of equivalence of measures plays a fundamental role in both probability theory and measure theory. It allows us to compare and relate different ways of assigning probabilities or measuring sets, enabling us to switch between different measures while preserving key properties. This concept finds widespread applications in various areas, including probability calculations, integration, and the study of stochastic processes. Understanding the significance and applications of equivalence of measures is essential for a comprehensive understanding of these mathematical fields.
