Let $M$ be the Sigma algebra of measurable sets, then the Lebesgue measure m(E) of any set $E∈M$ is________.

• A) $μ(E)$
• B) a complex valued function
• C) $m^{∗}(A)$
• D) a real valued function

Intervals in R are always_______.

• A) closed
• B) non-measurable
• C) open
• D) measurable

For given $∅≠E⊆R$ and $ε>0$, there exists a closed set $F$________.

• A) $F-E≠ empty~ set$
• B) F=E
• C) $F⊇E$
• D) $F⊆E$

$${\text{The sequence of intervals }}\left( { - \infty ,b - \frac{1}{n}} \right]{\text{ (as }}n \to \infty ){\text{ converges to which of the following sets?}}$$

• A) $$\left( { - \infty ,b - 1} \right)$$
• B) $$\left( { - \infty ,b - 1} \right]$$
• C) $$\left( { - \infty ,b} \right]$$
• D) $$\left( { - \infty ,b} \right)$$

If the interval $\left( -\infty ,a\right)$ is measurable in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ and $A\subseteq %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ is a test set, then $m^{\ast }\left\{ A\cap \left( a,\infty \right) \right\} +m^{\ast }\left\{ A\cap \left( -\infty ,a\right) \right\}$ ____________.

• A) $\leq m^{\ast }\left( -\infty ,a\right)$
• B) $=m^{\ast }\left( -\infty ,a\right)$
• C) $=m^{\ast }\left( a,\infty \right)$
• D) $=m^{\ast }\left( A\right)$

An open set U in R is measurable, because it can be expressed as________.

• A) countable union of open sets
• B) arbitrary union of open sets
• C) arbitrary intersection of open sets
• D) countable intersection of open sets

$${\text{Which of the following statements is true if A}} \subset \bigcup\limits_{n = 1}^\infty {{I_n}} {\text{?}}$$

• A) $$\sum\limits_{n = 1}^\infty {l({I_n})} = {m^*}(A) + \varepsilon$$
• B) $$\sum\limits_{n = 1}^\infty {l({I_n})} > {m^*}(A) + \varepsilon$$
• C) $$\sum\limits_{n = 1}^\infty {l({I_n})} < {m^*}(A) + \varepsilon$$
• D) $$\sum\limits_{n = 1}^\infty {l({I_n})} = {m^*}(A)$$

For all $E⊆R$, there is a $G_{δ}$ set $G⊇E$, such that_______.

• A) $m*(G-E)>0$
• B) $m*(G-E)$ is not defined
• C) $m*(G-E)<0$
• D) $m*(G-E)=0$

For all $E⊆R$, there is a $F_{σ}$ set $F⊆E$, such that_______.

• A) $m*(E-F)$ is not defined
• B) $m*(E-F)=0$
• C) $m*(E-F)<0$
• D) $m*(E-F)>0$

For given $∅≠E⊆R$ and $ε>0$, there exists an open set $O$________.

• A) $O⊇E$
• B) $E⊆O$
• C) $E-O≠∅$
• D) $E=O$

If the non-empty sets say $A,B$ are measurable then $A-B$__________.

• A) may or may not measurable
• B) implies $m^{∗}(A-B)<ε$
• C) is also measurable
• D) is not measurable

If a set in Euclidean space is translated by a fixed amount in any direction, what happens to its Lebesgue measure?

• A) It decreases.
• B) It remains the same.
• C) It becomes infinite.
• D) It increases.

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

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

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

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

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

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

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

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

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

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

Those countable sets which are Borel have _________ measure.

• A) positive
• B) infinite
• C) no
• D) zero

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

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

What property does the Cantor set possess despite containing an uncountable infinity of points?

• A) Finite length
• B) Uncountable length
• C) Infinite length
• D) Zero length