Sigma math n1/1/2024 ![]() ![]() If anyone could provide a more intuitive explanation or even better give an example (finite, like a die roll), I would be very grateful. I get that we have the power set of Omega that definitely contains the collection A - But what exactly do we mean by intersecting all the sigma algebras containing A to find the smallest one containing A? Does it mean that if we have a sigma algebra containing the collection A and another collection of subsets, B (which is a sigma algebra containing A, but i get that it is not the smallest) and intersect it with the power set of Omega, we generate sigma(A), which is indeed the smallest and more refined to answer the questions that we need in our problem? But, where exactly does the bigger sigma algebra (on the collections A and B) come from? The last part is the one I don't understand and confuses me. Further, we can find the smallest sigma algebra by intersecting all sigma algebras containing A, as the intersection of sigma algebras is also a sigma algebra. I do understand the definition Let A be an arbitrary collection of subsets of Omega, then sigma(A) is the generated sigma algebra, generated from A and is the smallest sigma algebra containing A. My problem is with generated sigma algebras. ![]() In addition I know that we have the trivial sigma algebra, the smallest sigma algebra on Omega and the Discrete Sigma Algebra, which is the power set of Omega, being the largest sigma algebra on Omega. So far so good and I also understand the properties that derive from the definition as well as how they are derived. if we have a collection of events) denoted by F, then F is a sigma-algebra if it satisfies the following three conditions While I do understand the concept if we have a set which is a collection of subsets of Omega (i.e. So I am sorry if this is trivial to most of you and apologies for any conceptual mistakes I might make in the description - I'll try to be as precise as possible.Īt the moment, I am studying Probability Theory, from this course:, , and I stumbled upon sigma algebras. ![]() I am not a mathematician, rather I pick up on topics on the go, when I need something for the topic I am studying in the given time. ![]()
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