We define measurable functions, give examples and establish equivalent conditions for a function on a measurable space to be measurable. We then end the session with the algebra of measurable functions.
Timestamp provided by Kogulraj K.
00:00 – Intro
01:47 – Definition of real-valued measurable functions
02:57 – P.R.P. 1
03:24 – Example 1: Constant functions
04:32 – Example 2: Indicator functions
08:36 – P.R.P. 2
08:44 – Example 3: Continuous functions
10:14 – P.R.P. 3
10:20 – Example 4: Monotone functions
14:18 – P.R.P. 4
14:44 – Example 5: Simple functions
17:49 – P.R.P. 5
18:07 – Definition of extended real-valued measurable functions
20:00 – Example 6: Non example
22:40 – P.R.P. 6
22:47 – Precap
23:38 – Standard definition of measurable functions
24:36 – Theorem: Equivalent conditions of measurable functions
26:43 – Idea for the proof
27:38 – Proof of (i) implies (ii)
28:59 – P.R.P. 7
29:05 – Proof of (ii) implies (iii)
30:37 – P.R.P. 8
30:44 – Proof of (iii) implies (i)
35:00 – Proof of (iv) & (v)
35:43 – P.R.P. 9
35:50 – Algebra of measurable functions
36:33 – f and g measurable implies { f greater than g } is measurable
39:53 – P.R.P 10
40:07 – f is measurable and c is real implies f + c is measurable
40:37 – f is measurable and c is real implies cf is measurable
41:19 – f and g are r. v. measurable fn. implies f + g is a r. v. measurable fn.
42:30 – P.R.P. 11
42:50 – f and g are e. r. v. measurable fn. implies f + g is a e. r. v. measurable fn.
43:46 – f is measurable and g is continuous implies g o f is measurable
48:23 – P.R.P. 12
48:30 – f is measurable implies f_square is measurable
48:58 – P.R.P. 13
49:15 – f is measurable implies |f| is measurable
49:51 – f and g are measurable implies max{f,g} and min{f,g} are measurable
52:28 – Some more applications of the lemma
53:00 – P.R.P. 14
53:03 – f and g are measurable implies fg is measurable
53:52 – f is nonzero and measurable implies 1/f is measurable
55:25 – P.R.P. 15
55:33 – Outro
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