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

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

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

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

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

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

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

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

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

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

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

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

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

\[{\text{Which of the following sets is a Borel set?}}\]

• A) \[{F_\sigma }{\text{ set}}\]
• B) \[{\text{The set of rational numbers}}\]
• C) \[{\text{All of these}}\]
• D) \[{G_\delta }{\text{ set}}\]

A set E is a Borel set if it can be expressed as_____________.

• A) countable union of open (closed) sets
• B) countable intersection of open (closed) sets
• C) neither countable union nor intersection of open or closed sets
• D) either countable union or intersection of open or closed sets

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

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

\[{\text{Let }}f:\mathbb{R} \to \mathbb{R}{\text{ defined by }}f(x) = 2.{\text{ The set }}\{ x \in \mathbb{R}:f(x) > 7\} {\text{ is equal to which of the following sets?}}\]

• A) \[\mathbb{R}\]
• B) \[\left( {7,\infty } \right)\]
• C) \[\emptyset \]
• D) \[\left( { - \infty ,7} \right)\]

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

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

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 $\varnothing \neq E\in M$ is measurable if the set $\left\{ x\in E:f(x)>a\right\} $ is measurable for all __________.

• A) $x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \cup \left\{ \pm \infty \right\} $
• B) $a\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \cup \left\{ \pm \infty \right\} $
• C) $a\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $
• D) $x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $

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

{\text{Which of the following sets is a Borel set?}}

• A) {F_\sigma }{\text{ set}}
• B) {\text{All of these}}
• C) {\text{The set of rational numbers}}
• D) {G_\delta }{\text{ set}}

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

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

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

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

• 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) \[{\text{A function that preserves the order of elements in its domain}}{\text{.}}\]

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

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