A(an)__________of Lebesgue measure $m$ to the class of Borel sets $% \mathit{B}$ is said to be Borel measure space.

• A) restriction
• B) kernel
• C) extension
• D) bijection

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

What distinguishes the Cantor set from intervals of positive length?

• A) Its self-similarity
• B) Its infinite cardinality
• C) Its measure being zero
• D) Its uncountable nature

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

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

Which of the following is a property of Borel sets?

• A) Borel sets are always finite.
• B) Borel sets cannot contain uncountably many elements.
• C) Every Borel set is measurable.
• D) The complement of a Borel set is always an open set.

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

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

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

• A) 5
• B) 3
• C) 4
• D) 2

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

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

According to George Cantor;

• A) Cardinality$(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )\geq$Cardinality$(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n})$
• B) $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion =% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}$
• C) Cardinality$(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )=$Cardinality$(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n})$
• D) Cardinality$(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )\leq$Cardinality$(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n})$

$${\text{Which operation generates }}{G_\delta }{\text{ Borel sets from open sets?}}$$

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

$${\text{A singleton set }}\{ x\} {\text{ can be written as:}}$$

• A) $$\bigcap\limits_{n = 1}^\infty {\left( {x - \frac{1}{n},{\text{ }}x + \frac{1}{n}} \right)}$$
• B) $$\bigcup\limits_{n = 1}^\infty {\left( {\frac{1}{n},{\text{ }}\frac{1}{n}} \right)}$$
• C) $$\bigcap\limits_{n = 1}^\infty {\left( { - \frac{1}{n},{\text{ }}\frac{1}{n}} \right)}$$
• D) $$\bigcup\limits_{n = 1}^\infty {\left( {x - \frac{1}{n},{\text{ }}x + \frac{1}{n}} \right)}$$

For a non-empty set $E$ in a measurable set $M$________.

• A) $m(E)=m^{\ast }(E)$
• B) $m(E)>m^{\ast }(E)$
• C) $m(E)<m^{\ast }(E)$
• D) $m(E)=2m^{\ast }(E)$

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

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

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

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

Suppose $f$ is a real valued identically constant function $f(x)=\beta ,$ then $\left\{ x:f(x)>\alpha \right\} =% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$. And if $\beta >\alpha$, then__________.

• A) $f$ is $1-1$
• B) $f~$is not measurable
• C) $f$ is bijective
• D) $f$ is measurable

Let $M$ be a measurable space (a sigma algebra of measurable sets) then a function $f:E\longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$ on $\varnothing \neq E\in M$ given as $\left\{ x\in E:f(x)>a\right\}$ is measurable if $\forall a,$----------under $f$ is measurable set.

• A) inverse image of $\left( a,\infty \right)$
• B) image of $\left( x,\infty \right)$
• C) inverse image of $\left( a,\infty \right)$
• D) image of $\left( a,\infty \right)$

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

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

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

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

What is a characteristic function in mathematics?

• A) A function that describes the characteristics of a set
• B) A function that calculates the average of a set
• C) A function that assigns values to elements of a set
• D) A function that indicates membership of an element in a set

What does the characteristic function XA∩B​(x) represent?

• A) The maximum value of the characteristic functions of sets A and B.
• B) The minimum value of the characteristic functions of sets A and B.
• C) The difference of the characteristic functions of sets A and B.
• D) The sum of the characteristic functions of sets A and B.