Recent content by ryo0071

[FONT=arial]Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...

Sorry about that. We have that $$A, B \subset \mathbb{R}^n$$. We developed measure has follows: Let $$a, b \in \mathbb{R}^n$$. A special rectangle is $$I = \{x \in \mathbb{R}^n | a_i \leq x_i \leq b_i $$ for $$1 \leq i \leq n\}$$ The measure of $$I$$ is $$\lambda(I) = (b_1 - a_1)\cdots(b_n -...

Let $$\lambda(A)$$ denote the measure of $$A$$ and let $$\lambda^{*}(A)$$ denote the outer measure of $$A$$ and let $$\lambda_{*}(A)$$ denote the inner measure of $$A$$ Okay so the question is as follows: Suppose that $$A \cup B$$ is measurable and that $$\lambda(A \cup B) = \lambda^{*}(A) +...

Thank you for your response. I probably should have mentioned I have taken care of the cases where $$x_1 = 0$$ and $$x_2 \not= 0$$ as well as $$x_1 \not= 0$$ and $$x_2 = 0$$. Also, I am aware that it would be continuous since it is the result of operations of continuous function but I am trying...

Okay so the question is: Let $$f:R^2 \rightarrow R$$ by $$f(x) = \frac{x_1^2x_2}{x_1^4+x_2^2}$$ for $$x \not= 0$$ Prove that for each $$x \in R$$, $$f(tx)$$ is a continuous function of $$t \in R$$ ($$R$$ is the real numbers, I'm not sure how to get it to look right). I am letting $$t_0...

Thank you both for the help. I was trying to use the hint but what was confusing me was that they used the same variable x for both parts of the change of coordinates (rather than doing something like $$x \rightarrow \eta^2$$). Anyway, I was able to solve it by method of characteristics without...

Okay so I am working on this problem: Solve $$xu_t + uu_x = 0$$ with $$u(x, 0) = x.$$(Hint: Change variables $$x \rightarrow x^2$$.) However, I am not sure how to use the change of variables hint that is given or why it is needed. My thinking is that I could just use the method of...

Thanks for the links. I look forward to taking a course on functional analysis.

Thanks for the quick replies. I figured it probably would be defined like that. How would one test for convergence? (I suppose a better question is how does one define a norm for a linear operator?) And another question would be what part of math would you study things like these (functions of...

In class we recently learned that for a linear operator $$T: V \rightarrow V$$ and function $$g(t) = a_0 + a_1t + \dots + a_nt^n$$ one can define the operator $$g(T) = a_0I + a_1T + \dots + a_nT^n$$ (where $$I$$ is the identity transformation). We also recently learned about the exponential of...