Multiple choice questions on Measure Theory

1. The outer measure of a non-empty set A is denoted by______.

a. m(A)

b. \[\overset{.}{\mathop{m}}\,(A)\]

c.\[\overset{\_}{\mathop{m}}\,A\]

d. \[\underset{\_}{\mathop{m}}\,(A)\]

2. The inner measure of a non-empty set A is denoted by_____.

a.\[ \underset{\_}{\mathop{m}}\,(A)\]

b. \[\overset{\_}{\mathop{m}}\,A\]

c. m(A)

 

3. If A = [2,3] and B = [0,4] then m(A)______m(B). The blank space is____.

a. >

b. <

c. =

d. None of the above

 

4. If A₁ and A₂ are measureable subsets of [a, b] then A₁UA₂ is_______.

a. Measurable set

b. Integrable set

c. Non-Measurable set

d. None of the above

 

5. If A₁ and A₂  are measurable subsets of [a, b] then  A₁ - A₂  is______.

a. Measurable set

b. Integrable set

c. Non-Measurable set

d. None of the above

 

6. If A₁ and A₂  are measurable subsets of [a, b] then  A₁⋂A₂ is______.

a. Measurable set

b. Integrable set

c. Non-Measurable set

d. None of the above

 

7. If A₁ and A₂  are measurable subsets of [a, b] then  A₁ ∆ A₂  is______.

a. Measurable set

b. Integrable set

c. Non-Measurable set

d. None of the above

 

8. A countable set is______.

a. Finite non-measurable set

b. Integrable set

c. Infinite non-measurable set

d. Measurable

 

9. if \[\overset{\_}{\mathop{m}}\,(E)\]=0 then E is______.

a. Measurable set

b. Non measurable set

c. Intergrable set

d. None of the above

 

10. A set containing of one point is______ and its measure is_______.

a. Measurable, 0

b. Non-measurable, 0

c. Measurable, 1

d. Non-measurable, 1

 

11. For any set A, _ m A x (A)______( ) +. Fill in the blanks.

a. <

b. >

c. =

d. ≠

 

12.  There exists an uncountable set of measure. The set is called _____.

a. Cauchy set

b. Cantor set

c. Riemann set

d. Lebesgue set

 

13. If f is a measurable function, then |f| is______.

a. Measurable function

b. Non-measurable function

c. Constant function

d. Modulus function

 

14. If f is a measurable function, then the set {x/f(x)=s} is …….

a. Non-Measurable set

b. Measurable set

c. Equal set

d. None of the above

 

15. If f1 and f2 are measurable functions then on [a, b] then\[ \frac{{{f}_{1}}}{{{f}_{2}}}\]  is measurable on [a, b] provided that f2 ______.

a. =0

 b. ∞

c. ≠0

d. −∞

 

16.  If f is a measurable function, then the set {x/f(x)=s} is_____.

a. Non-Measurable set

b. Measurable set

c. Equal set

d. None of the above

 

17. If f1 and f2 are measurable functions then on [a, b] then f1 + f2 is ______on [a, b]

a. Measurable function

b. Non-measurable function

c. Summable function

d. None of the above

 

18. If f1 and f2 are measurable functions then on [a, b] then f1 - f2 is_____on [a, b]

a. Non-Measurable function

b. Measurable function

c. Not defined

d. None of the above

 

19. If f1 and f2 are measurable functions then on [a, b] then f1 X f2 is ……… on [a, b]

a. Measurable function

b. Non-measurable function

c. Not defined

d. None of the above

 

20. Every continuous function is_______.

a. Non-Measurable function

b. derivable

c. Measurable function

d. integrable

 

21. F=g a.e. and f is a measurable function then g is also measurable. A.E. stands for ….

a. All except

b. Almost everywhere

c. All early

d. All equal

 

22. ⋃[P, f] is called _________

a. Upper Riemann sum

b. Upper Lebesgue sum

c. Lower Riemann sum

d. Lower Lebesgue sum

 

23. L [P, f] is called ________

a. Upper Riemann sum

b. Upper Lebesgue sum

c. Lower Riemann sum

d. Lower Lebesgue sum

 

24. The Cantor set C, ______

a. is countable

b. is uncountable

c. have measure 0

d. Both (b) and(c)

25. A subset G = [3,4] of an interval [1,4] is_______ in[1,4].

a. closed

b. open

c. neither open nor closed

d. either open or closed