Danny Pak-Keung Chan

 10 Spivak's Calculus: chapter 2, problem 18(c) 10 If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$ 8 $f_n \rightharpoonup f$, $g_n \to g$ in measure, $\|g_n\|_{L^\infty} \leq M$, then $f_n g_n \rightharpoonup fg$, 7 If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$? 7 Prove that $\binom{n}{0} ^2 + \binom{n}{1} ^2 + \binom{n}{2} ^2 + … + \binom{n}{n} ^2 = \binom{2n}{n}$.

### Reputation (7,921)

 +25 $g_{n}(x):=max \lbrace f_{1}(x),…\rbrace$ uniformly convergent if $\lbrace f_{n} \rbrace_{n \in \mathbb{N}}$ uniformly bounded and equicontinuous +10 $g_{n}(x):=max \lbrace f_{1}(x),…\rbrace$ uniformly convergent if $\lbrace f_{n} \rbrace_{n \in \mathbb{N}}$ uniformly bounded and equicontinuous +25 Random Variable convergence in $L^{q}$ space, then the conditional expectation also converges in $L^{q}$ +10 Assistance in showing the limit of $\lim_{(x,y) \to (0,0)} \frac{x \sin^{2}y}{x^{2} + y^{2}}$ exists.

### Questions (2)

 1 Well-definedness of a random variable $V\circ p$, P.86 of GTM 261 (Probability and Stochastics) 0 Non-negative r.v. $X_n$, with $X_n\rightarrow 0$ in probability and $E[X_n]\rightarrow 2$. Prove that $\lim_nE[|X_n-1|]$ exists and find the limit.

### Tags (168)

 233 real-analysis × 185 49 lebesgue-integral × 37 130 measure-theory × 101 48 sequences-and-series × 27 61 general-topology × 43 45 integration × 29 56 calculus × 36 38 convergence-divergence × 25 49 probability-theory × 48 37 functional-analysis × 40

### Bookmarks (1)

 10 For a differentiable function $f$ show that $\{x:\limsup_{y\to x}|f'(y)|<\infty\}$ is open and dense in $\mathbb R$

### Accounts (12)

 Mathematics 7,921 rep 88 silver badges1616 bronze badges Quantitative Finance 131 rep 22 bronze badges Cross Validated 111 rep 22 bronze badges MathOverflow 101 rep TeX - LaTeX 101 rep