![]() ![]() There are many phenomena that follow this basic “size”-like idea including such phenomena as mass, length, and volume (and as we will see later, probability). What do we mean by size? Size is a number that we attribute to an object that obeys a specific, intuitive property: if we break the object apart, the sizes of the pieces should add up to the size of the whole. Measure theory is all about abstracting the idea of “size”. #Unintuitive phenomena definition full$\sigma$-algebra? Measure space? Let’s introduce each of these ideas, and then we will come back to this full three-part definition. The measure $P$ for the measureable space $(\Omega, E)$ is the probability measure.The $\sigma$-algebra over $\Omega$, denoted $E$, called the set of events.The set $\Omega$, is called the sample space.First, the formal definition of a probability space:ĭefinition 1: A probability space is a measure space ($\Omega$, $E$, $P$) where $P(\Omega) = 1$ where I’ll introduce a bunch of terms in my definition for probability and in the following sections, we will define and discuss each of them. A measure theoretic foundation for probability ![]() Sadly, this bridge comes at the cost of a less satisfying, and more tedious explanation of probability. It is no wonder why teachers build a bridge to cross this crevice and avoid measure theory altogether. In fact, I was surprised to learn that the very formal definition of probability requires measure theory! It is as if the very first step into probability plunges one into a crevice of mathematics. I learned that this division, between continuous and discrete distributions, is actually a bridge over a deeper and more complete theory of probability, based on measure theory, that not only unifies the discrete and continuous scenarios, but also extends them. Not only did I find this division to be unsatisfying, but as I continued to study statistics and machine learning through grad school, I found it to be inadequate for a deeper understanding into the workings of the topics that I was studying. Whereas if $X$ is continuous its expectation is given by\ For example, the expectation of a discrete random variable $X$ is given by.\ a dice roll) are treated quite differently from continuous outcomes (e.g. IntroductionĪs a computer science undergraduate student, I was taught the basic and incomplete version of probability theory in which discrete outcomes (e.g. #Unintuitive phenomena definition seriesIn this series of posts, I will present my understanding of some basic concepts in measure theory - the mathematical study of objects with “size”- that have enabled me to gain a deeper understanding into the foundations of probability theory. Demystifying measure-theoretic probability theory (part 1: probability spaces) ![]()
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