illysial

### Questions (52)

 18 The closure of a connected set in a topological space is connected 11 Will the Lebesgue integral of a real valued function always be a Riemann sum? 7 If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous. 6 A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable 5 Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

### Reputation (1,688)

 +5 Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable +5 If $f_n$ is integrable and non-negative, $f_n \rightarrow f$ a.e, $\int f_n \rightarrow \int f$. prove $\int_A f_n \rightarrow \int_A f$ +5 If $\mu$ is $\sigma$-finite, then there exist increasing simple functions $s_n \rightarrow f$ with $\mu(\{x:s_n \neq 0\})< \infty$ +10 How to find the period of the sum of two trigonometric functions

 7 Is there an easier way to find $\frac{\mathrm d^9}{\mathrm dx^9}(x^8\ln x)$ than using the product rule repeatedly? 4 Solve the initial value problem: $\frac{dx}{dt}=2t \sin x$; $x(0)=\frac{\pi}{2}$ 3 find a limit of $\lim_{n\rightarrow\infty}\left(\frac{a^{1/n}+b^{1/n}}{2}\right)^n$ 3 The limit of $e^{\frac{1}{x^2 + y^2}}$ as $(x, y) \to (0, 0)$ 3 using newtons method to find abs max

### Tags (67)

 27 calculus × 20 5 functions × 3 15 derivatives × 5 4 abstract-algebra × 4 11 multivariable-calculus × 8 4 ordinary-differential-equations 8 linear-algebra × 11 3 measure-theory × 21 8 limits × 3 3 real-analysis × 12