The document discusses measurable functions, defining them as extended real-valued functions on measurable sets for which certain conditions involving measurable sets are equivalent. It presents proofs for theoretical statements about constant functions and demonstrates the properties of measurable functions through various theorems, including cases of restrictions to measurable subsets and continuity. Additionally, references are provided by Dr. Sunil K. Mittal and Dr. Sudhir K. Punidir.

1. CHAUDHARY BANSILAL
UNIVERSITY,BHIWANI
Topic:- Properties of measurable
function
SUBMITTED BY:- SARITA
ROLL NO:-220000603043
CLASS:-M.SC 1ST 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)>𝜶} 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) >𝛼} is
measurable for each number
Now
{x∈E:f(x) >𝛼} = E , k>𝛼
𝜑,k= 𝛼
𝜑, 𝑘 < 𝛼
Since both the set E and𝜑 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 𝛼,{x∈E : f(x)>𝛼} is measurable.
(ii)for each real number 𝛼,{x∈E : f(x)≥ 𝛼} is measurable.
(iii)for each real number 𝛼,{x∈E : f(x)< 𝛼} is measurable.
(iv) For each real number 𝛼,{x∈E : f(x)≤ 𝛼} is measurable.

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

6. Similarly, we can prove (iii) ⇒ (𝑖𝑣) and (ii) ⇒ (𝑖𝑖𝑖)
{x:f(x) <𝛼}=E- {x: f(x) ≥ 𝛼}
Since E is measurable by (ii)
{x : f(x)≥ 𝛼} is measurable and since difference
of two measurable set is measurable.
{x : f(x)<𝛼} is measurable
Now we prove (i) ⇒ (ii)
{x:f(x) ≥}= 𝑛=1
∞
{x: f(x) >𝛼 −
1
𝑛
{
So {x: f(x) >𝛼 −
1
𝑛
{is measurable.
Since countable intersection of measurable is
measurable.
𝑛=1
∞
{x: f(x) >𝛼 −
1
𝑛
{is measurable
Hence {x : f(x)≥ 𝛼} is measurable
Hence (i) ⇒ (ii)

7. Properties of measurable functions
Theroem
(a) If f is a measurable function on the set E and 𝐸1 ⊂ E is measurable set , then f is
a measurable function 𝐸1
Proof:-
For each real 𝛼 , we have 𝐸1 (f>𝛼) = 𝐸(𝑓 > 𝛼) ∩ 𝐸1.
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
{𝐸𝑖} 𝑜𝑓 𝑑𝑖𝑠𝑗𝑜𝑖𝑛𝑡 measurable sets , then f is a measurable on 𝑖 𝐸𝑖
Proof:-
Write E= 𝑖=1
∞
𝐸𝑖 clearly , E is measurable set (A countable union of measurable sets is
a measurable sets) .
Since for each real 𝛼 , 𝑤𝑒 ℎ𝑎𝑣𝑒
E(f>𝛼) = 𝑖=1
∞
𝐸𝑖 (𝑓 > 𝛼).

9.  If f is a measurable function on a measurable set A and B ⊂ 𝐴 𝑖𝑠 𝑎 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