suppose s is the smallest algebra on r containing r r 1 r q prove that s is the collection of bo

Step 1: The goal is to show that the collection S is the collection of Borel subsets of R. The B

Suppose s is the smallest algebra on r containing r r 1 r q...

Transcript

00:01 Okay, so the question is as follows.

00:04 We look at the set s, which we say is the smallest sigma algebra.

00:10 Where is the sigma algebra generated by the sigma algebra in r, right? generated by the set r comma r plus 1, where r is a rational number.

00:34 So this is the problem statement.

00:39 It should be on r.

00:41 Okay, so moreover, we need to show that s is the collection of borrell subsets of r.

00:52 So we need to say that s is the borel subsets on r.

01:00 Okay.

01:03 Well, the borrell set on r is, you can think of it at the smallest sigma algebra generated by, let's say, a basis for the topology of and a basis for the topology of r is this is a set a, b where a and b are elements of the reels.

01:31 So the topology, the standard topology on r is generated by such sets.

01:39 And if we can show that s contains a set of the form a comma b where a and b are real numbers, then that's sufficient to demonstrate that s is, well, the borel sigma algebra on r.

02:00 And the way that we do this is, well, we grab one of these, the goal here is to make the set a, b, right? where a and b are real numbers, using, or four, from sets of the form r comma r plus one, where r is a rational, taking these sets to, these such sets to be an element of a sigma algebra.

02:50 Okay, then the way that we do this is, let me make some space for this...

Question

Suppose S is the smallest $sigma$-algebra on $mathbb{R}$ containing ${(r, r + 1) : r in mathbb{Q}}$. Prove that S is the collection of Borel subsets of $mathbb{R}$.

Question

Suppose S is the smallest $sigma$-algebra on $mathbb{R}$ containing ${(r, r + 1) : r in mathbb{Q}}$. Prove that S is the collection of Borel subsets of $mathbb{R}$.

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