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

• A) 0
• B) 2
• C) -1
• D) 1

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

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

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

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

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

{\text{Let }}f:\mathbb{R} \to \mathbb{R}{\text{ defined by }}f(x) = 9.{\text{ The set }}\{ x \in \mathbb{R}:f(x) > 7\} {\text{ is equal to which of the following sets?}}

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

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

{\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 $