Suppose $f$ is a real valued identically constant function $f(x)=\beta ,$ then $\left\{ x:f(x)>\alpha \right\} =$__________, if $\beta =\alpha$.

• A) $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$
• B) $\left\{ \beta \right\}$
• C) $\left\{ \alpha \right\}$
• D) $\varnothing$

What does monotonicity in characteristic functions refer to?

• A) The function's behavior as the input varies.
• B) The function's relationship with set inclusion.
• C) The function's derivative being constant.
• D) The function's continuity over its domain.

In $\left[ 0,1\right) ,$ the sum modulo $1,x\widehat{+}y$ of $\frac{1}{2}$ and $\frac{1}{2}=$_________.

• A) $0$
• B) $-1$
• C) $2$
• D) $1$

If~\{E_{n}\}~be~an~increasing~sequence~of~measurable~sets~then

• A) ...E_{3}\subseteq E_{2} \subseteq E_{1} \subseteq
• B) E_{1}-E_{1}\subseteq E_{1}-E_{2}\subseteq E_{1}-E_{3}\subseteq... E_{1}-E_{n}\subseteq....
• C) E_{1}\neq E_{2}\neq E_{1}...
• D) E_{1}\subseteq E_{2} \subseteq E_{3} \subseteq ...

In the first iteration for constructing Cantor Set from [0,1], we divide it in_____________sub-intervals.

• A) 5
• B) 4
• C) 2
• D) 3

The Lebesgue outer measure of a Cantor set constructed from [0,1], is________________.

• A) \frac{1}{3}
• B) 0
• C) \infty
• D) 1

If \left\{ E_{n}\right\} be a decreasing sequence of measurable sets then____________.

• A) E_{1}\neq E_{2}\neq E_{3}
• B) E_{1}-E_{1}\subseteq E_{1}-E_{2}\subseteq E_{1}-E_{3}\subseteq \cdots E_{1}-E_{n}\subseteq \cdots
• C) E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \cdots
• D) E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \cdots

If \left\{ E_{n}\right\} be a sequence of measurable sets such that % E_{1}=E_{2}=E_{3}\cdots then \left\{ E_{n}\right\} is an example of____________sequence.

• A) increasing
• B) both increasing and decreasing
• C) neither increasing nor decreasing
• D) decreasing

In the first iteration for constructing Cantor Set from [0,1], we drop a sub-interval of length__________.

• A) 1
• B) \frac{1}{3}
• C) \frac{1}{4}
• D) \frac{1}{2}

In the 2nd Iteration for constructing Cantor Set from [0,1],we drop a sub-intervals of length ________________.

• A) \frac{1}{3^{2}}
• B) \frac{1}{3^{0}}
• C) \frac{1}{3^{1}}
• D) \frac{1}{3^{3}}

The Cantor set constructed from [0,1] is ________________.

• A) finitely countable
• B) uncountable
• C) infinitely countable
• D) finite

Let \varnothing \neq A,B\in \mathbf{M:}the measurable space, such that B\subseteq A, then ________.

• A) m(B)\geq m(A)
• B) m(B)\leq m(A)
• C) m(B-A)=m(A)
• D) m(B)=m(A)

Let \varnothing \neq A\in \mathbf{M:}the measurable space, such that t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion , then________.

• A) m(A\times t)=m(A+t)
• B) m(A\times t)=m(A\div t)
• C) m(A\div t)=m(A-t)
• D) m(A-t)=m(A)

If $\left\{ E_{n}\right\} $ be a decreasing sequence of measurable sets then____________.

• A) $E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \cdots $
• B) $E_{1}-E_{1}\subseteq E_{1}-E_{2}\subseteq E_{1}-E_{3}\subseteq \cdots E_{1}-E_{n}\subseteq \cdots $
• C) $E_{1}\neq E_{2}\neq E_{3}$
• D) $E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \cdots $

If $\left\{ E_{n}\right\} $ be a sequence of measurable sets such that $% E_{1}=E_{2}=E_{3}\cdots $ then $\left\{ E_{n}\right\} $ is an example of____________sequence.

• A) neither increasing nor decreasing
• B) decreasing
• C) increasing
• D) both increasing and decreasing

{\text{Which of the following best describes a measurable function?}}

• A) {\text{A function that maps Borel sets to Lebesgue measurable sets}}{\text{.}}
• B) {\text{A function that can be represented as the union of countably many open sets}}{\text{.}}
• C) {\text{A function that maps measurable sets to measurable sets}}{\text{.}}
• D) {\text{A function that preserves the order of elements in its domain}}{\text{.}}

Let x,y\in \left[ 0,1\right) ,then the sum modulo 1;x\widehat{+}y of \ x and \ y is defined by_________.

• A) x+y if x+y\leq 1 and x+y-1 if x+y>1
• B) x+y if x+y<1 and x+y-1 if x+y\geq 1
• C) x+y-1
• D) x+y if x+y\geq 1 and x+y-1 if x+y<1

{\text{Which of the following sets is countable and also a Borel set with measure zero?}}

• A) {\text{The interval [0,1]}}
• B) {\text{The set of natural numbers}}
• C) {\text{The set of irrational numbers}}
• D) {\text{The set of real numbers}}

In \left[ 0,1\right) , the sum modulo 1,x\widehat{+}y of \frac{3}{2} and \frac{1}{2}=___________.

• A) 3
• B) is not defined at all
• C) \frac{1}{2}
• D) 1

{\text{Which operation generates }}{F_\sigma }{\text{ Borel sets from closed sets?}}

• A) Countable complementation
• B) Countable negation
• C) Countable union
• D) Countable intersection