Definitionand properties of measurable function.pptx

definition of measurable function
some theorem
properties

1. CHAUDHARY BANSILAL UNIVERSITY,BHIWANI Topic:- Properties of measurable function submitted by:- sarita roll no:-220000603043 class:- m.Sc 1 st year (maths department)

2. MEASURABLE FUNCTIONS An extended real valued function f defined on a measurable set E is said to be measurable function if {x E: f(x)> is measurable for each real number.  

3. Theorem:- A constant function with a measurable domain is measurable. Proof:- Let f be constant function with measurable domain E and Let f:E(measurable) R be a constant function i.e., f(x)= k , where k is constant. We have to show that {x E : f(x) > } is measurable for each number Now {x E:f(x) > } = E , k> ,k= Since both the set E and are measurable  

4. Theorem:- Let f be an extended real valued function defined on measurable set E then the following conditions are equivalent. ( i )for each real number ,{x E : f(x)> } is measurable. (ii)for each real number ,{x E : f(x) } is measurable. (iii) for each real number ,{x E : f(x) } is measurable. (iv) For each real number ,{x E : f(x) } is measurable.  

5. Proof :-We show that these four conditions are equivalent. Let E is measurable ( i ) (iv) E:f(x) }= E- {x E : f(x)> } Since E is measurable and by (i) {x E:f(x) > } is measurable. Also E is measurable and difference of two measurable set is measurable. {x: f(x) } is measurable. (iv) :f(x) }= E- {x: f(x) } {x: f(x) } is measurable and difference of two measurable set is measurable. {x E : f(x)> } is measurable  

6. Similarly, we can prove (iii) and (ii) {x:f(x) < }=E- {x: f(x) } Since E is measurable by (ii) {x : f(x) } is measurable and since difference of two measurable set is measurable. {x : f(x)< } is measurable Now we prove (i) (ii) {x:f(x) }= x: f(x) > } So {x: f(x) > } is measurable. Since countable intersection of measurable is measurable. x: f(x) > } is measurable Hence {x : f(x) } is measurable Hence (i) (ii)  

7. Properties of measurable functions Theroem If f is a measurable function on the set E and E is measurable set , then f is a measurable function Proof:- For each real we have (f> . The set on the right hand side is measurable.  

8. (b) If f is a measurable function on each of the set in a countable collection { measurable sets , then f is a measurable on Proof:- Write E= clearly , E is measurable set (A countable union of measurable sets is a measurable sets) . Since for each real E(f> .  

9. If f is a measurable function on a measurable set A and B measurable set on B. If f is continuous function defined on set E which is a measurable set , then f is measurable function. A continuous function on a closed interval is measurable.  

10. REFRENCE BY Dr.Sunil k. Mittal Dr.Sudhir K. Punidir

11. THANK YOU