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Tarea 15, Análisis real

Due November 20

Problem 1

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

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

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

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

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

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

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

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

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

Problem 2

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

Problem 3

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

Problem 4

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

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

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

Problem 5

For each $latex A\subset\R$ define

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

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