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## Due November 20

### Problem 1

$\mathcal B_\R$ is generated by each of the following:

1. $\mathcal E_1 = \{(a,b): a,b\in\R, a<b\}$

2. $\mathcal E_2 = \{[a,b]: a,b\in\R, a<b\}$

3. $\mathcal E_3 = \{(a,b]: a,b\in\R, a<b\}$

4. $\mathcal E_4 = \{[a,b): a,b\in\R, a<b\}$

5. $\mathcal E_5 = \{(a,\infty): a\in\R\}$

6. $\mathcal E_6 = \{(-\infty,a): a\in\R\}$

7. $\mathcal E_7 = \{[a,\infty): a\in\R\}$

8. $\mathcal E_8 = \{(-\infty,a]: a\in\R\}$

### Problem 2

An algebra $\sigma$-algebra iff it is closed under countable increasing unions.

### Problem 3

If $\mu_1, \ldots,\mu_n$ are measures on $(X,\mathcal M)$ and $a_1,\ldots,a_n\ge0$, then $\sum_{j=1}^n a_j\mu_j$ is a measure on $(X,\mathcal M)$.

### Problem 4

Let $(X,\mathcal M,\mu)$ be a measure space and $E_i\in\mathcal M$.

1. $\mu(\liminf E_i) \le \liminf \mu(E_i)$.

2. If $\mu(\cup_i E_i)<\infty$, then $\mu(\limsup E_i) \ge \limsup \mu(E_i)$.

### Problem 5

For each $A\subset\R$ define

$\displaystyle m^*(A) = \inf \Big\{\sum_{i=1}^\infty (b_i - a_i): A\subset\bigcup_{i=1}^\infty (a_i,b_i)\Big\}$.

Then $\displaystyle m^*\Big(\bigcup_{j=1}^\infty A_j\Big) \le \sum_{j=1}^\infty m^*(A_j)$.