Measurable Spaces – Problem (46/365)

The rest of the chapter on \sigma-algebras goes through the construction of various other measurable spaces such as those on the space of 1) continuous functions, 2) functions continuous on the right, and 3) direct products of measurable spaces. The next chapter introduces methods of introducing probability measures on measurable spaces.

The way we do this is to start with a distribution function F(x) from which we derive a unique probability measure where P(a,b]) = F(b) - F(a). Here is a problem.

Let F(x) = P(-\infty, x), then verify that P(a,b] = F(b) - F(a).

\displaystyle  P(a,b] = P\left( (-\infty,b] \setminus (-\infty, a] \right) \\ = P(-\infty, b] - P(-\infty, a] \text{ by additivity} \\ = F(b) - F(a)

Verify that P(a,b) = F(b-) - F(a) where F(x-) = \lim_{y \uparrow x} F(y).

\displaystyle  P(a,b) = P \left( \cup_{n=1}^\infty P(a,b-\frac{1}{n}] \right) \\ = \lim_n P(a,b - \frac{1}{n}] \text{ because } P \text{ is countably additive over } \mathcal{B}(R) \\ = \lim_n F(b - \frac{1}{n}) - F(a) \\ = F(b-) - F(a)

The proof for the following are similar: P[a,b] = F(b) - F(a-), P[a,b) = F(b-) - F(a-), and P\{x\} = F(x) - F(x-).

Advertisements
This entry was posted in Uncategorized and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s