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Thanks for the response. Could you say whether for finite sets, my statement is correct? Also could you explain when measures and quantity are the same and wh

by j7ake 3y ago

Thanks for the response. Could you say whether for finite sets, my statement is correct? Also could you explain when measures and quantity are the same and when they are not?

aquafox 3y ago
TL;DR #1 is correct. #2: Quantity usually means just a number but isn't a maths term. A measures is a function that maps sets to numbers. A probability is a special kind of measure. For finite sets it is correct. More specifically, the quantity you are talking about is the number 1 and it is distributed over all elements of the set such that every element gets a number between 0 and 1 (where also the borders 0 or 1 are allowed), and the numbers of all elements sum up to 1. Quantity is not a mathematical term, but we usually use it in every day language to just denote a number. A measure, mathematically, is a function. A measure of a set A assigns _every subset_ of A to a number greater or equal 0 and has two key properties: 1) The empty set is always mapped to the number 0. 2) The union of every two subsets X and Y of A, that also don't have any element in common, is mapped to the sum of the values of the subsets X and Y, i.e. measure(X ∪ Y) = measure(X) + measure(Y). The cool thing is: Probabilities are a special kind of measures! Remember that a measure has to assign a number for every subset of a set A? Since A is a subset of itself, a measure has to also assign A a value. A so-called probability-measure is a measure that assigns A the value 1. Now you might guess where this is going: If you want to define probabilities for something going on in the real world, you first define a set that contains all the things that can happen. E.g. if you define probabilities for a dice roll, the set A contains the 6 possible numbers, A = {1,2,3,4,5,6}. Each subset of A corresponds to a possible event. For example, the set {2,5} means "either 2 or 5 appears". How is the probability-measure P defined? If the dice is fair, we have P({1})=P({2})=P({3})=P({4})=P({5})=P({6})=1/6. From property 2), we then get P({2,5})=1/6+1/6=1/3, i.e. the probability that 2 or 5 appear is 1/3. And of course we have P({1,2,3,4,5,6}) = 1, i.e. the probability that any number appears is 1. Sorry for the long reply, but I hope I could clarify a few things. Maths gets very fast very technical, but it's also nice to understand some key concepts.

by j7ake 3y ago

Thanks for the response. Could you say whether for finite sets, my statement is correct? Also could you explain when measures and quantity are the same and when they are not?

aquafox 3y ago

TL;DR #1 is correct. #2: Quantity usually means just a number but isn't a maths term. A measures is a function that maps sets to numbers. A probability is a special kind of measure. For finite sets it is correct. More specifically, the quantity you are talking about is the number 1 and it is distributed over all elements of the set such that every element gets a number between 0 and 1 (where also the borders 0 or 1 are allowed), and the numbers of all elements sum up to 1. Quantity is not a mathematical term, but we usually use it in every day language to just denote a number. A measure, mathematically, is a function. A measure of a set A assigns _every subset_ of A to a number greater or equal 0 and has two key properties: 1) The empty set is always mapped to the number 0. 2) The union of every two subsets X and Y of A, that also don't have any element in common, is mapped to the sum of the values of the subsets X and Y, i.e. measure(X ∪ Y) = measure(X) + measure(Y). The cool thing is: Probabilities are a special kind of measures! Remember that a measure has to assign a number for every subset of a set A? Since A is a subset of itself, a measure has to also assign A a value. A so-called probability-measure is a measure that assigns A the value 1. Now you might guess where this is going: If you want to define probabilities for something going on in the real world, you first define a set that contains all the things that can happen. E.g. if you define probabilities for a dice roll, the set A contains the 6 possible numbers, A = {1,2,3,4,5,6}. Each subset of A corresponds to a possible event. For example, the set {2,5} means "either 2 or 5 appears". How is the probability-measure P defined? If the dice is fair, we have P({1})=P({2})=P({3})=P({4})=P({5})=P({6})=1/6. From property 2), we then get P({2,5})=1/6+1/6=1/3, i.e. the probability that 2 or 5 appear is 1/3. And of course we have P({1,2,3,4,5,6}) = 1, i.e. the probability that any number appears is 1. Sorry for the long reply, but I hope I could clarify a few things. Maths gets very fast very technical, but it's also nice to understand some key concepts.