5. Describe the ?-algebra F of subsets of ? generated by the...

5. Describe the ?-algebra F of subsets of ? generated by the singleton sets (i.e., sets of the form {a} where a is any real number.) • Does F contain intervals of the form (a, b) for b > a? • Define a probability measure on F. • Describe the random variables that can be defined on the space (?, F). We learned about random variables in class and you worked with them in the last example. Note that whether or not a function X : ? ? ? is a random variable depends on the ?-algebra F of the probability space (?, F, ?)! Definition. For a function X : ? ? ? define ?(X) to be the smallest ?-algebra which makes X measurable (and therefore a random variable) with respect to ?(X). We read it as “the sigma algebra generated by X”. Why does it exist? Definition. For random variables X, Y we say that ‘‘X is Y-measurable’’ if X is measurable with respect to ?(Y).

Transcript

00:01 Hi there, welcome to number one.

00:03 We're going to cover the question relating to measure theory.

00:06 So you have a sigma a n -zebra f, and you have to show that if they not contain any intervals, right? okay, so how do we do that? so the question here is no, right? then how to prove it? so first let me remind you what is the definition of sigma -denegra is generated by a set, right? so, so first let me remind you what is the definition of sigma -0 -0 -0 -0 -0 -0 -0 -0.

00:32 So, so first let me remind you what is the definition of the definition of the so if i have a set a and i have a sigma an zebra, let denote it by sigma a, which means that it's the smallest sigma angera containing a.

00:46 And now we have to prove that this does not contain any interval.

00:54 So i denote the set, let's see, g is another sigma angela, right? on r such that so e is measurable if and only e is countable or ec is countable right so so so you can take g is actually a sigma a zera right and you can because the set at a sigma and zva containing all the singleton sets right so because it's it's close in the countable union so which means that any countable set belongs to f right in all the word we have g is containing a g contains all the single single single single sets right and g is a sigma and so g is a bigger set than f because f is the sigma a zebra can generated by singleton sets so it's the smallest is sigma angela so f is a subset of t and now you have to rule that g is a subset of f right so if you pick up a set chee uh e belong to chi right so e is measurable or ec is a measurable sorry is a countable set or is it is a countable set if e is a countable set if it is countable which means that it is equal to some countable union of x -i right so which means that e belong to f by definition if e is not countable then ec is countable right and then you do the same thing right ec equal to this.

03:12 So, which means that ec belongs to f and because it closed under the complementary, right? so, e also belongs to x and therefore you rule the statement...

Question

Please give Ace some feedback