We often have the condition of "compact support" in theorems. Why is it important? Are there any special properties relies on it? For example, the random variable is bounded if its distribution has compact support.
I have two questions here: 1. why is the random variable bounded if its distribution has compact support? 2.Are there any other porperties relied on the compactness of support like this? Why we often have to add this condition?
This is my first time to ask questions. Thank you very much!
Recall that a compact on the real numbers is close and bounded. If the distribution support of a r.v. is bounded, this implies that outside a bounded set the random variable takes values with probability zero. Equivalently, it is bounded with probability 1. However, the importance of the compact support depends of the particular theorem you are interested in, this is not strictly necessary and there are a myriad of commonly used and important r.v. without compact support.
I am not sure what you mean by "compactness of the support of a measure". But maybe checking some of the properties of compact spaces (here) would help you understanding why is it required for your specific case.
As this is your first question I would also recommend you to have a look at this to improve your chances to get an answer on what you need.
Hope this helps.