Right, but I didn't entirely understand how this pertained to the question of whether \int_{0}^{\infty}x^{n}f(x)\mathrm dx = 0 for each integer n >= 0 implies that f(x) = 0 for x nonnegative (which is what my attempted proof used, perhaps incorrectly referring to those integrals as "moments")...
Thanks, but I'm not sure I understand. I'm looking at the integral from 0 to infinity of x^n*f(x). In this case, it would be \int_{0}^{\infty}x^{n}e^{\frac{-1}{x^{2}}} \mathrm dx , which is infinite for each n rather than 0.
I realize now that I may have been misusing the term "moment;" I was...
Hello, I was trying to prove that the Laplace transform is unique and was wondering if anyone could tell me if I've made any errors in my attempt. Here it is:
Suppose L(f) = L(g), where L() denotes the Laplace transform. We want to show that f = g. By linearity of the transform, L(f - g) = 0...
O.K., here's an easy counterexample to the general claim:
\displaystyle \sum_{i=1}^{\infty} \lim_{n\rightarrow\infty} \frac{1}{2^{n}} = 0, \ \lim_{n\rightarrow\infty} \sum_{i=1}^{\infty} \frac{1}{2^{n}} = \infty.
Limits usually behave so naturally in terms of commuting with other...
I understand that the limit of the sum of two sequences equals the sum of the sequences' limis: \displaysyle \lim_{n\rightarrow\infty} (a_{n} + b_{n}) = \lim_{n\rightarrow\infty}a_{n} + \lim_{n\rightarrow\infty}b_{n}. Similar results consequenly hold for sums of three sequences, four sequences...
Hi, I'm new here. I'm trying to teach myself measure theory and probability and recently wanted to find an example of a probability space (X, A, P) where X is uncountable and the sigma-algebra A is the entire power set of X. Here was my idea: let X be the set of all strings c1c2c3... for c_{i}...