Real Analysis II Question Bank

Which of the following is not measurable function?

• Constant function

• Continuous function

• Characteristic function

• For sequence of measurable functions {fn}, supfn

If f(x) is any real function, then f-x= . . . . . . . . . . . . .

Say True or False: Every Borel set is a Measurable set.

Show that the class of measurable sets is a σ-algebra.

For any set A there exists a measurable set E containing A and such that m*A=mE.

Let c be any real number and let f and g be real-valued measurable functions defined on the same measurable set E. Then show that f+c, cf, f+g, f-g and fg are measurable.

Is every measurable set a Borel set? Justify your answer.

(i) For any sequence of sets Ei, prove that m*(i=1∞Ei)≤i=1∞m*Ei.

If f is an integrable function, then fdx≤fdx. When does equality holds?

• f≥0 a.e

• f≤0 a.e

• both (a) and (b)

• neither (a) nor (b)

When we say that a function is periodic function?