What in the AHA do you measure with Measure Theory? | Multiverse Vistas 2026

[00:00:08]Hi there, teacher Aanasa here. Welcome to today's super special Evan measure ti class. As the ruler of the phantoism moon games when it comes to measuring things, I am undoubtedly the expert. I mean, I do have this certified ruler of mine after all. Apparently, Aha thought it would be funny if I taught you a little something about measure. Now, you might be wondering, what in the aha do you measure with measure theory? Is it length, area, or maybe volumes? Trivia.

[00:00:43]Way too trivial. Don't you agree, Vubu?

[00:00:48]Today, let's learn how to measure even the most ridiculous math objects imaginable, whether they are finite or infinite. Countable or uncountable.

[00:00:59]But before we start, we have to start at the very start. A set theory.

[00:01:09]What is a set in math? A set is a collection of objects. For example, I want to make a set of the fluffy across the blue characters. So this set has elements baseball racing.

[00:01:27]This type of set is called a finite countable set. Another type of set is an infinite countable set. For example, the set of natural numbers 1 2 3 4 and so on. With this infinite life of mine, I can count the elements of an infinite countable set. We also have uncountable sets. These are usually in the form of intervals of real numbers. For example, take the closed interval of 0 and one.

[00:01:56]Let's count together 0.1.

[00:02:03]No, wait. 0.01.

[00:02:06]What about 0.001?

[00:02:10]The point is, even with my infinite lifespan, you still can't count the elements of an uncountable set.

[00:02:20]Let us define a universal set as omega which contains all the elements we're interested in. In that case, let's have our fluffy universe as our omega.

[00:02:31]Let us have set a like this and set B like this both being subsets of omega.

[00:02:37]Now we shall learn all kinds of set operations.

[00:02:41]First, the complement of set A is a set of all elements not in set A but still in omega.

[00:02:49]If all elements of A are also elements of B, we can call set A as a subset of B.

[00:02:56]If B is a subset of B accent, then B accent is a subset of B, then B and B accent are set to be equal. Now let's define new sets of A and B. The set of all elements which belong to either a B is called the set of A union B. The set of all elements which belong to both A and B is called the set of A intersection B. In particular, if the intersection of both sets is an empty set, then set A and B is said to be disjoint. Now, there is one important law in set theory. Team Oen's law. It may sound scary, but it's just a rule explaining how unions on intersections interact under compliments. Let's use our fluffy universe again. Suppose that set A is baseball raccoon and March Bunny while set B is March Bunny and Woof woof.

[00:03:48]Then I union contains everyone who isn't either A or B. Now take the compliment that is himat, Mr. Yang, and Sandalf.

[00:04:00]Notice that the complement of A is woof woof Dan Hikat Mr. Yang and Sund while the complement of B is baseball raccoon Hikat Mr. Yang and Sund. So their intersection where everyone who is neither in A nor in B is Mr. Yang and Sund.

[00:04:21]Did you see that? This matches our previous compliment of the union. Can you guess what that means? Mab boo.

[00:04:29]Yes.

[00:04:31]This is the first theme organ's law. The second law works similarly. Let's draw it.

[00:04:39]This is the intersection of A and B. The set where its element is not in both sets is equal to sets where its element is missing from at least one set.

[00:04:54]Now suppose we have a collection of sets. Nomega. Let's call it the class. A >> for example, we can make a class from subsets of our fluffy universe like so.

[00:05:06]>> Some of this is more special than the others.

[00:05:09]>> If we perform a certain operation on sets in a >> and the result is still an A, >> we say that A is closed under that operation.

[00:05:18]A field is a class that is closed under complementation then finite intersection.

[00:05:24]What's a finite intersection you ask?

[00:05:27]Basically you intersect the finite amount of sets in a >> what 2 3 4 5 6 or even 100 sets but still finite.

[00:05:38]>> Using mathematical induction we can prove that if closed under pair wise intersection then a >> is also closed under finite intersection.

[00:05:48]As a consequence a field is also closed under finite union thanks to D Morgan's law. Last but not least, every field also must contain the empty set omega.

[00:06:00]In fact, a class that only contains the empty set in omega is a field called the degenerate or trivial field.

[00:06:13]A sigma field is a special type of field. Basically, a field that is closed under countable intersections and countable unions is called a sigma field. By countable I mean both finite and infinite countable. Remember that the set of natural numbers is countable.

[00:06:32]So if we perform intersections of sets indexed from 1 to infinity that's infinitely many yet countable. The result must still belong to the sigma field.

[00:06:42]This is what closed under countable intersection means.

[00:06:46]Just because a class is closed under finite operations doesn't mean that it is also closed under countable operations.

[00:06:53]Can you think of an example? La boo boo.

[00:06:59]Okay, here's an example. Let a be the class of open intervals of the form 1 - 1 / n to the finite intersection of sets and a >> is also in the form of this open interval. Thus closed under finite intersection.

[00:07:16]Now if we try to perform countable intersection of sets belonging in a we obtain a left closed interval from 1 to two which by definition is not a set in a.

[00:07:28]>> Therefore we know that is not closed under countable intersection.

[00:07:33]A sigma field has all the properties of a field. In fact if there's a field that contains only infinitely many sets then a is automatically a sigma field too.

[00:07:45]Any set lucky enough to belong to a sigma field is called a measurable set. So what in the aha is the deal with measuring sets?

[00:07:58]>> Let be a sigma field generated by a class of sets.

[00:08:03]>> A measure is a map from a set to the real number.

[00:08:07]>> Of course, this map must satisfy some particular conditions. The measure of an empty set is by definition zero.

[00:08:15]Therefore, since any set is bigger than an empty set, the measure of any set belonging to A must be non- negative.

[00:08:23]And last, for any countable collections of pairwise disjoint sets of A, >> the measure of their union is equal to the sum of their individual measure. In other words, measuring them all at once or one by one then adding them together must give the same result.

[00:08:40]We can measure any measurable sets however we want. Size is relative after all. Example, for a countable finite set, its measure could be its cardality or the number of elements in the set.

[00:08:54]For an uncountable set like an interval in a real number, we can measure it by the length of its interval or the value of the integral of some function over the interval. If the set is an event, then its measure could be the probability of such an event occurring.

[00:09:13]All right, that marks the end for today's lesson. Next time, let's learn even more properties of fields, sigma fields, and measures, and perhaps how to construct them with me. Hope you learn something new.

[00:09:33]Hey, hey, hey.