We define sigma-algebras on a set and measures on sets with a fixed sigma algebra. We give four examples which will be used in the sessions on integration.
Timestamp by Kogularaj K:
00:50 – Intro
02:14 – Definition of a sigma algebra
03:50 – Some examples of sigma algebras
05:27 – Intersection of arbitrary collection of sigma algebras
08:09 – P.R.P. 1
08:32 – Smallest sigma algebra containing a family of subsets
11:13 – Borel sigma algebra on R^n
12:20 – Borel sets are measurable
14:04 – P.R.P. 2
14:14 – Recap
15:07 – Definition of a measurable space
17:08 – P.R.P. 3
17:17 – Some more examples of measurable spaces
18:20 – P.R.P. 4
18:36 – Definition of a measure on a measurable space
20:43 – P.R.P. 5
21:54 – Precap on examples of measures
21:33 – Example 1: The Lebesgue measure
22:04 – Definition of a measure space
22:52 – Example 2: The Borel measure
24:24 – Example 3: The Counting measure
27:15 – P.R.P. 6
27:32 – Example 4: The Counting measure with Weights
30:33 – Example 5: The Dirac measure
32:54 – Properties of a measure space
33:11 – Property 1: Finite additivity of a measure
35:56 – P.R.P. 7
36:02 – Property 2: Monotonicity of a measure
40:27 – Property 3: Countable subadditivity of a measure
43:07 – P.R.P. 8
43:11 – Outro
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