一〇二上實分析筆記@S-2013.09.11

下星期三作業繳至 Exercise 1.3.1.

Recall Theorem 1.1 Insert non-formatted text here .2
If $r_\alpha\geq0$ for all $\alpha$, then $\{r_\alpha\}_{\alpha\in I}$ is summable if and only if $$\sum_{\beta\in B}r_\beta$$ is bounded (above) among all $B\in\mathcal F(I)$. If it is the case, then the least upper bound is the desired sum.

§1.2 Double Series
Let index set $I$ be $\mathbb N\times\mathbb N$. Simply re-denote $r_{(i,j)}$ by $r_{ij}$. It is summable if it satisfies pervious definition.

Theorem 1.2.1
If the double series $\{r_{ij}\}_{ij}$ is summable, then $$\sum_Ir_{ij}=\lim_{m,n\to\infty}\sum_{i=0}^m\sum_{j=0}^nr_{ij}=\sum_i\sum_jr_{ij}=\sum_j\sum_ir_{ij}$$

proof. 我懶

Remark
Let $r_\alpha=r_\alpha^+-r_\alpha^-$, then $\{r_\alpha\}_{\alpha\in I}$ summable if and only if $\{r_\alpha^+\}_{\alpha\in I}$ and $\{r_\alpha^-\}_{\alpha\in I}$ are both summable. So that we may assume $r_{ij\geq0}$ above, and use "least upper bound" to claim the sum.

§1.3 Coin Tossing
Bernoulli Trial, with probability $p,q$ to be Head and Tail. Sample space is $\Omega=\{H,T\}^\infty$. An Elementary Cylinder is an event with the form $(\epsilon_1,\dotsc,\epsilon_n)\times\{H,T\}^{\infty-n}\subset\{H,T\}^\infty$, with rank $n$. A Cylinder is any finite union of such elementary cylinders.