UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
If $I$ is a finite nonempty set of sets in a semiring of sets $M$, then the intersection of the sets in $I$ is also in $M$.
If $x$ is a member of a semiring of sets $M$, then $\Omega \cap x = x$.
If $x$ is an element of a semiring of sets $M$, then $x \cap \Omega = x$.
If $P$ and $Q$ are measurable sets, then so is the set of points $x$ such that $P(x)$ and $Q(x)$ are both true.
If $S$ is a finite non-empty set of sets in a semiring of sets, then the intersection of all the sets in $S$ is also in the semiring.
If $X$ is a finite set of sets in a ring of sets $M$, then the union of $X$ is also in $M$.
If $I$ is a finite set and $A_i \in M$ for all $i \in I$, then $\bigcup_{i \in I} A_i \in M$.
If $a$ and $b$ are elements of a ring of sets $M$, then $a - b$ is also an element of $M$.
If $M$ is a collection of subsets of $\Omega$ that is closed under finite unions, finite intersections, and complements, then $M$ is a ring of sets.
A set $M$ is a ring of sets if and only if $M$ is a subset of the power set of $\Omega$, $M$ contains the empty set, $M$ is closed under union, and $M$ is closed under set difference.
If $x$ and $A$ are in $M$, then $x \cup A$ is in $M$.
If $P$ and $Q$ are sets in $M$, then $P \cup Q$ is also in $M$.
If $S$ is a finite set and $P_i$ is a property that holds for each $i \in S$, then the set of elements that satisfy at least one of the $P_i$ is in $M$.
If $a$ is an element of a Boolean algebra $M$, then $\Omega - a$ is also an element of $M$.
A set $M$ is an algebra if and only if it is a subset of the power set of $\Omega$, contains the empty set, is closed under complements, and is closed under unions.
A set $M$ is an algebra if and only if $M$ is a subset of the power set of $\Omega$, $\Omega \in M$, $\Omega - a \in M$ for all $a \in M$, and $a \cap b \in M$ for all $a, b \in M$.
If $M$ is a $\sigma$-algebra and $P$ is a predicate, then the set $\{x \in \Omega \mid \lnot P(x)\}$ is in $M$.
If $P$ and $Q$ are sets in a sigma algebra, then so is the set of all $x$ such that $Q(x) \implies P(x)$.
If $P$ is a property of elements of $\Omega$, then the set of elements of $\Omega$ that satisfy $P$ is in $M$.
If $X$ is a subset of $S$, then the algebra generated by $X$ is the set of all subsets of $S$.
If $A$ is a subalgebra of a module $M$, then $A$ is an algebra over the restricted space of $M$.
If $M$ is a finite set of subsets of $\Omega$, then $\sigma(M)$ is a $\sigma$-algebra.
If $A$ is a countable family of sets, then $\bigcup_{i \in I} A_i$ is the same as $\bigcup_{i \in \mathbb{N}} A_{f(i)}$ for any bijection $f: \mathbb{N} \to I$.
If $X$ is a countable subset of a $\sigma$-algebra $M$, then $\bigcup X \in M$.
If $A$ is a countable collection of sets in a $\sigma$-algebra $M$, then the union of the sets in $A$ is also in $M$.
If $X$ is a countable set and $A$ is a function from $X$ to a $\sigma$-algebra $M$, then the union of the sets $A(x)$ for $x \<in> X$ is in $M$.
If $X$ is a countable set and $A$ is a function from $X$ to $M$, then $\bigcup_{x \in X} A(x)$ is in $M$.
If $A$ is a countable collection of sets in a $\sigma$-algebra $M$, then the intersection of all the sets in $A$ is also in $M$.
If $X$ is a countable set of sets in a $\sigma$-algebra $M$, then the intersection of all the sets in $X$ is also in $M$.
If $M$ is a $\sigma$-algebra, $I$ is a countable set, and $F_i \in M$ for all $i \in I$, then $\bigcap_{i \in I} F_i \in M$.
If $A$ is a countable set of sets in a $\sigma$-algebra $M$, then $A$ is in $M$.
The power set of a set is a ring of sets.
The power set of a set is an algebra.
A collection of subsets of a set $\Omega$ is a $\sigma$-algebra if and only if it is an algebra and the countable union of any sequence of sets in the collection is also in the collection.
The power set of a set $S$ is a $\sigma$-algebra.
If $P_i$ is a countable collection of measurable sets, then $\bigcap_{i=1}^\infty P_i$ is measurable.
If $P_i$ is a countable collection of sets in a $\sigma$-algebra, then the union of the $P_i$ is also in the $\sigma$-algebra.
If $I$ is a countable set and $P_i$ is a property for each $i \in I$, then the set of all $x$ such that $P_i(x)$ holds for some $i \in I$ is in the sigma algebra.
If $I$ is a countable set and $P_i$ is a property for each $i \in I$, then the set of all elements $x$ such that $P_i(x)$ holds for all $i \in I$ is measurable.
If $I$ is a countable index set and $P_i$ is a family of sets in a $\sigma$-algebra $M$, then the set of points $x$ such that there is exactly one $i \in I$ with $P_i(x)$ is in $M$.
The following lemmas are useful for proving that a set is measurable.
If $P_i$ is a $\sigma$-algebra for each $i$, then $\{x \in \Omega \mid \forall i \in X, P_i(x)\}$ is a $\sigma$-algebra.
If $P_i$ is a countable collection of sets in a $\sigma$-algebra, then the union of the $P_i$ is also in the $\sigma$-algebra.
If $X \subseteq S$, then the collection of sets $\{ \emptyset, X, S - X, S \}$ is a $\sigma$-algebra on $S$.
The range of the binary function is the set $\{a,b\}$.
The union of two sets is the union of all binary representations of the sets.
The intersection of two sets $a$ and $b$ is the set of all elements that are in both $a$ and $b$.
A set $M$ is a $\sigma$-algebra if and only if $M$ is a subset of the power set of $\Omega$, $\emptyset \in M$, $\Omega - s \in M$ for all $s \in M$, and $\bigcup_{i=1}^{\infty} A_i \in M$ for all sequences $A_1, A_2, \ldots$ of sets in $M$.
If $a$ is a subset of a $\sigma$-algebra $M$, then the $\sigma$-algebra generated by $a$ is a subset of $M$.
If $A$ is a subset of the power set of $X$, and $x$ is in the $\sigma$-algebra generated by $A$, then $x$ is a subset of $X$.
If $a$ is a subset of the power set of $\Omega$, then the sigma-algebra generated by $a$ is a sigma-algebra.
The $\sigma$-algebra generated by a collection of sets is the intersection of all $\sigma$-algebras containing that collection.
The topology on a space is a $\sigma$-algebra.
If $a$ and $b$ are in $\sigma$-algebras $A$ and $B$, then the binary operation $a \cup b$ is in the $\sigma$-algebra generated by $A$ and $B$.
If $a$ and $b$ are $\sigma$-sets, then $a \cup b$ is a $\sigma$-set.
If $A$ is a subset of the power set of a set $S$, and $a_i$ is a sequence of elements of the $\sigma$-algebra generated by $A$, then the intersection of the $a_i$ is also an element of the $\sigma$-algebra generated by $A$.
If $A$ is a subset of the power set of a set $S$, and $a_i$ is a $\sigma$-algebra on $S$ for each $i \in S$, then the intersection of all the $a_i$ is a $\sigma$-algebra on $S$.
If $B$ is a countable set of sets in $\sigma(A)$, then $\bigcup B \in \sigma(A)$.
The $\sigma$-algebra generated by a set $M$ is equal to $M$.
If $A$ and $B$ are two sets of subsets of a measurable space $(X, \mathcal{M})$ such that every element of $A$ is a $\sigma$-set and every element of $B$ is a $\sigma$-set, then $A$ and $B$ are equal.
If $A \subseteq B$, then $\sigma(A) \subseteq \sigma(B)$.
If $A$ is a subset of the $\sigma$-algebra generated by $B$, then the $\sigma$-algebra generated by $A$ is a subset of the $\sigma$-algebra generated by $B$.
If $A \subseteq B$, then $\sigma(A) \subseteq \sigma(B)$.
The sigma-algebra generated by a set $A$ is a superset of $A$.
If $S$ is a measurable set and $A_i$ is a sequence of measurable sets such that $A_i \cap S$ is measurable for all $i$, then $A_i$ is measurable for all $i$ and $\bigcup_i A_i$ is measurable.
If $S$ is a measurable set, then the $\sigma$-algebra of the restricted space $S$ is the $\sigma$-algebra of $S$.
If $A$ is a $\sigma$-algebra on a set $X$, then the $\sigma$-algebra generated by the intersection of $A$ with any subset of $X$ is the same as the intersection of $A$ with the $\sigma$-algebra generated by that subset.
The $\sigma$-algebra generated by the empty set is the set of all subsets of $A$ and the empty set.
The $\sigma$-algebra generated by a single set $A$ is $\{\emptyset, A\}$.
If $M$ is a subset of the power set of $S$, then the $\sigma$-algebra generated by $M$ is equal to the $\sigma$-algebra generated by the $\sigma$-algebra generated by $M$.
If $X$ is a subset of $S$, then the $\sigma$-algebra generated by $X$ is the set of all subsets of $S$ that are either empty, equal to $X$, equal to $S - X$, or equal to $S$.
If $S$ is a $\sigma$-algebra on $\Omega$ and $M$ is a $\sigma$-algebra on $\Omega$, then the $\sigma$-algebra on $S$ generated by the $\sigma$-algebra on $S$ generated by $M$ is the same as the $\sigma$-algebra on $S$ generated by the $\sigma$-algebra on $S$ generated by $M$.
If $X$ is a measurable function from $\Omega$ to $\Omega'$, then the preimage of a $\sigma$-algebra on $\Omega'$ is a $\sigma$-algebra on $\Omega$.
If $A_0, A_1, \ldots, A_{n-1}$ are sets in a ring of sets $M$, then $\bigcup_{i=0}^{n-1} A_i$ is also in $M$.
If $A$ is a collection of subsets of a set $X$, then the range of the collection of disjoint sets of $A$ is a subset of $X$.
If $A$ is a set of matrices with entries in a ring $R$, then the set of matrices obtained by applying the disjoint function to each matrix in $A$ is also a set of matrices with entries in $R$.
A collection of sets is a $\sigma$-algebra if and only if it is an algebra and the union of any disjoint family of sets in the collection is also in the collection.
If $a$ is an element of the generated ring, then there exists a finite set $C$ of mutually disjoint elements of $M$ such that $a = \bigcup C$.
If $C$ is a finite collection of disjoint sets in $M$, then $\bigcup C$ is in the generated ring.
If $A$ is a member of a semiring of sets $M$, then $A$ is a member of the generated ring of $M$.
If $a$ and $b$ are in the generated ring and $a \cap b = \emptyset$, then $a \cup b$ is in the generated ring.
The empty set is in the generated ring.
If $A$ is a finite set of disjoint sets, and each set in $A$ is in the generated ring, then the union of the sets in $A$ is in the generated ring.
If $A_1, \ldots, A_n$ are disjoint sets in a semiring of sets, then $\bigcup_{i=1}^n A_i$ is also in the semiring.
If $a$ and $b$ are elements of the generated ring, then $a \cap b$ is also an element of the generated ring.
If $A$ is a finite nonempty set of sets in a semiring of sets, then the intersection of the sets in $A$ is also in the semiring of sets.
If $I$ is a finite nonempty set of sets in a semiring of sets, then the intersection of the sets in $I$ is also in the semiring of sets.
The ring of sets of a semiring of sets is a generating ring.
The $\sigma$-algebra generated by a ring of sets is equal to the $\sigma$-algebra generated by the minimal ring of sets containing that ring.
The range of the binaryset function is $\{A,B,\emptyset\}$.
The union of all binary sets is equal to the union of the two sets.
The smallest closed countably determined intersection of sets containing $M$ is contained in the set of all closed countably determined intersections of sets containing $M$.
The smallest closed convex sets containing a set $M$ are a subset of the power set of $\Omega$.
The smallest closed convex decomposition of $\Omega$ is a closed convex decomposition of $\Omega$.
If $M$ is a closed subset of the power set of $\Omega$, then $M$ is a subset of the power set of $\Omega$.
If $\Omega$ is a closed convex set and $M$ is a closed convex cone, then $\Omega - s \in M$ for all $s \in M$.
If $A$ is a chain of sets in a monotone class $\Omega$ and $\Omega$ is closed under countable unions, then $\bigcup A \in \Omega$.
If $\Omega$ is a closed countably decomposable ideal, $M$ is a $\sigma$-algebra, $A$ is a disjoint family of sets in $M$, and the range of $A$ is contained in $M$, then the union of the sets in $A$ is in $M$.
If $M$ is a closed cover of $\Omega$ and $A, B \in M$ are disjoint, then $A \cup B \in M$.
If $A$ and $B$ are disjoint sets in the smallest $\sigma$-algebra containing all the cylinder sets, then $A \cup B$ is also in the smallest $\sigma$-algebra containing all the cylinder sets.

If $I$ is a finite nonempty set of sets in a semiring of sets $M$, then the intersection of the sets in $I$ is also in $M$.

If $x$ is a member of a semiring of sets $M$, then $\Omega \cap x = x$.

If $x$ is an element of a semiring of sets $M$, then $x \cap \Omega = x$.

If $P$ and $Q$ are measurable sets, then so is the set of points $x$ such that $P(x)$ and $Q(x)$ are both true.

If $S$ is a finite non-empty set of sets in a semiring of sets, then the intersection of all the sets in $S$ is also in the semiring.

If $X$ is a finite set of sets in a ring of sets $M$, then the union of $X$ is also in $M$.

If $I$ is a finite set and $A_i \in M$ for all $i \in I$, then $\bigcup_{i \in I} A_i \in M$.

If $a$ and $b$ are elements of a ring of sets $M$, then $a - b$ is also an element of $M$.

If $M$ is a collection of subsets of $\Omega$ that is closed under finite unions, finite intersections, and complements, then $M$ is a ring of sets.

A set $M$ is a ring of sets if and only if $M$ is a subset of the power set of $\Omega$, $M$ contains the empty set, $M$ is closed under union, and $M$ is closed under set difference.

If $x$ and $A$ are in $M$, then $x \cup A$ is in $M$.

If $P$ and $Q$ are sets in $M$, then $P \cup Q$ is also in $M$.

If $S$ is a finite set and $P_i$ is a property that holds for each $i \in S$, then the set of elements that satisfy at least one of the $P_i$ is in $M$.

If $a$ is an element of a Boolean algebra $M$, then $\Omega - a$ is also an element of $M$.

A set $M$ is an algebra if and only if it is a subset of the power set of $\Omega$, contains the empty set, is closed under complements, and is closed under unions.

A set $M$ is an algebra if and only if $M$ is a subset of the power set of $\Omega$, $\Omega \in M$, $\Omega - a \in M$ for all $a \in M$, and $a \cap b \in M$ for all $a, b \in M$.

If $M$ is a $\sigma$-algebra and $P$ is a predicate, then the set $\{x \in \Omega \mid \lnot P(x)\}$ is in $M$.

If $P$ and $Q$ are sets in a sigma algebra, then so is the set of all $x$ such that $Q(x) \implies P(x)$.

If $P$ is a property of elements of $\Omega$, then the set of elements of $\Omega$ that satisfy $P$ is in $M$.

If $X$ is a subset of $S$, then the algebra generated by $X$ is the set of all subsets of $S$.

If $A$ is a subalgebra of a module $M$, then $A$ is an algebra over the restricted space of $M$.

If $M$ is a finite set of subsets of $\Omega$, then $\sigma(M)$ is a $\sigma$-algebra.

If $A$ is a countable family of sets, then $\bigcup_{i \in I} A_i$ is the same as $\bigcup_{i \in \mathbb{N}} A_{f(i)}$ for any bijection $f: \mathbb{N} \to I$.

If $X$ is a countable subset of a $\sigma$-algebra $M$, then $\bigcup X \in M$.

If $A$ is a countable collection of sets in a $\sigma$-algebra $M$, then the union of the sets in $A$ is also in $M$.

If $X$ is a countable set and $A$ is a function from $X$ to a $\sigma$-algebra $M$, then the union of the sets $A(x)$ for $x \<in> X$ is in $M$.

If $X$ is a countable set and $A$ is a function from $X$ to $M$, then $\bigcup_{x \in X} A(x)$ is in $M$.

If $A$ is a countable collection of sets in a $\sigma$-algebra $M$, then the intersection of all the sets in $A$ is also in $M$.

If $X$ is a countable set of sets in a $\sigma$-algebra $M$, then the intersection of all the sets in $X$ is also in $M$.

If $M$ is a $\sigma$-algebra, $I$ is a countable set, and $F_i \in M$ for all $i \in I$, then $\bigcap_{i \in I} F_i \in M$.

If $A$ is a countable set of sets in a $\sigma$-algebra $M$, then $A$ is in $M$.

The power set of a set is a ring of sets.

The power set of a set is an algebra.

A collection of subsets of a set $\Omega$ is a $\sigma$-algebra if and only if it is an algebra and the countable union of any sequence of sets in the collection is also in the collection.

The power set of a set $S$ is a $\sigma$-algebra.

If $P_i$ is a countable collection of measurable sets, then $\bigcap_{i=1}^\infty P_i$ is measurable.

If $P_i$ is a countable collection of sets in a $\sigma$-algebra, then the union of the $P_i$ is also in the $\sigma$-algebra.

If $I$ is a countable set and $P_i$ is a property for each $i \in I$, then the set of all $x$ such that $P_i(x)$ holds for some $i \in I$ is in the sigma algebra.

If $I$ is a countable set and $P_i$ is a property for each $i \in I$, then the set of all elements $x$ such that $P_i(x)$ holds for all $i \in I$ is measurable.

If $I$ is a countable index set and $P_i$ is a family of sets in a $\sigma$-algebra $M$, then the set of points $x$ such that there is exactly one $i \in I$ with $P_i(x)$ is in $M$.

The following lemmas are useful for proving that a set is measurable.

If $P_i$ is a $\sigma$-algebra for each $i$, then $\{x \in \Omega \mid \forall i \in X, P_i(x)\}$ is a $\sigma$-algebra.

If $P_i$ is a countable collection of sets in a $\sigma$-algebra, then the union of the $P_i$ is also in the $\sigma$-algebra.

If $X \subseteq S$, then the collection of sets $\{ \emptyset, X, S - X, S \}$ is a $\sigma$-algebra on $S$.

The range of the binary function is the set $\{a,b\}$.

The union of two sets is the union of all binary representations of the sets.

The intersection of two sets $a$ and $b$ is the set of all elements that are in both $a$ and $b$.

A set $M$ is a $\sigma$-algebra if and only if $M$ is a subset of the power set of $\Omega$, $\emptyset \in M$, $\Omega - s \in M$ for all $s \in M$, and $\bigcup_{i=1}^{\infty} A_i \in M$ for all sequences $A_1, A_2, \ldots$ of sets in $M$.

If $a$ is a subset of a $\sigma$-algebra $M$, then the $\sigma$-algebra generated by $a$ is a subset of $M$.

If $A$ is a subset of the power set of $X$, and $x$ is in the $\sigma$-algebra generated by $A$, then $x$ is a subset of $X$.

If $a$ is a subset of the power set of $\Omega$, then the sigma-algebra generated by $a$ is a sigma-algebra.

The $\sigma$-algebra generated by a collection of sets is the intersection of all $\sigma$-algebras containing that collection.

The topology on a space is a $\sigma$-algebra.

If $a$ and $b$ are in $\sigma$-algebras $A$ and $B$, then the binary operation $a \cup b$ is in the $\sigma$-algebra generated by $A$ and $B$.

If $a$ and $b$ are $\sigma$-sets, then $a \cup b$ is a $\sigma$-set.

If $A$ is a subset of the power set of a set $S$, and $a_i$ is a sequence of elements of the $\sigma$-algebra generated by $A$, then the intersection of the $a_i$ is also an element of the $\sigma$-algebra generated by $A$.

If $A$ is a subset of the power set of a set $S$, and $a_i$ is a $\sigma$-algebra on $S$ for each $i \in S$, then the intersection of all the $a_i$ is a $\sigma$-algebra on $S$.

If $B$ is a countable set of sets in $\sigma(A)$, then $\bigcup B \in \sigma(A)$.

The $\sigma$-algebra generated by a set $M$ is equal to $M$.

If $A$ and $B$ are two sets of subsets of a measurable space $(X, \mathcal{M})$ such that every element of $A$ is a $\sigma$-set and every element of $B$ is a $\sigma$-set, then $A$ and $B$ are equal.

If $A \subseteq B$, then $\sigma(A) \subseteq \sigma(B)$.

If $A$ is a subset of the $\sigma$-algebra generated by $B$, then the $\sigma$-algebra generated by $A$ is a subset of the $\sigma$-algebra generated by $B$.

If $A \subseteq B$, then $\sigma(A) \subseteq \sigma(B)$.

The sigma-algebra generated by a set $A$ is a superset of $A$.

If $S$ is a measurable set and $A_i$ is a sequence of measurable sets such that $A_i \cap S$ is measurable for all $i$, then $A_i$ is measurable for all $i$ and $\bigcup_i A_i$ is measurable.

If $S$ is a measurable set, then the $\sigma$-algebra of the restricted space $S$ is the $\sigma$-algebra of $S$.

If $A$ is a $\sigma$-algebra on a set $X$, then the $\sigma$-algebra generated by the intersection of $A$ with any subset of $X$ is the same as the intersection of $A$ with the $\sigma$-algebra generated by that subset.

The $\sigma$-algebra generated by the empty set is the set of all subsets of $A$ and the empty set.

The $\sigma$-algebra generated by a single set $A$ is $\{\emptyset, A\}$.

If $M$ is a subset of the power set of $S$, then the $\sigma$-algebra generated by $M$ is equal to the $\sigma$-algebra generated by the $\sigma$-algebra generated by $M$.

If $X$ is a subset of $S$, then the $\sigma$-algebra generated by $X$ is the set of all subsets of $S$ that are either empty, equal to $X$, equal to $S - X$, or equal to $S$.

If $S$ is a $\sigma$-algebra on $\Omega$ and $M$ is a $\sigma$-algebra on $\Omega$, then the $\sigma$-algebra on $S$ generated by the $\sigma$-algebra on $S$ generated by $M$ is the same as the $\sigma$-algebra on $S$ generated by the $\sigma$-algebra on $S$ generated by $M$.

If $X$ is a measurable function from $\Omega$ to $\Omega'$, then the preimage of a $\sigma$-algebra on $\Omega'$ is a $\sigma$-algebra on $\Omega$.

If $A_0, A_1, \ldots, A_{n-1}$ are sets in a ring of sets $M$, then $\bigcup_{i=0}^{n-1} A_i$ is also in $M$.

If $A$ is a collection of subsets of a set $X$, then the range of the collection of disjoint sets of $A$ is a subset of $X$.

If $A$ is a set of matrices with entries in a ring $R$, then the set of matrices obtained by applying the disjoint function to each matrix in $A$ is also a set of matrices with entries in $R$.

A collection of sets is a $\sigma$-algebra if and only if it is an algebra and the union of any disjoint family of sets in the collection is also in the collection.

If $a$ is an element of the generated ring, then there exists a finite set $C$ of mutually disjoint elements of $M$ such that $a = \bigcup C$.

If $C$ is a finite collection of disjoint sets in $M$, then $\bigcup C$ is in the generated ring.

If $A$ is a member of a semiring of sets $M$, then $A$ is a member of the generated ring of $M$.

If $a$ and $b$ are in the generated ring and $a \cap b = \emptyset$, then $a \cup b$ is in the generated ring.

The empty set is in the generated ring.

If $A$ is a finite set of disjoint sets, and each set in $A$ is in the generated ring, then the union of the sets in $A$ is in the generated ring.

If $A_1, \ldots, A_n$ are disjoint sets in a semiring of sets, then $\bigcup_{i=1}^n A_i$ is also in the semiring.

If $a$ and $b$ are elements of the generated ring, then $a \cap b$ is also an element of the generated ring.

If $A$ is a finite nonempty set of sets in a semiring of sets, then the intersection of the sets in $A$ is also in the semiring of sets.

If $I$ is a finite nonempty set of sets in a semiring of sets, then the intersection of the sets in $I$ is also in the semiring of sets.

The ring of sets of a semiring of sets is a generating ring.

The $\sigma$-algebra generated by a ring of sets is equal to the $\sigma$-algebra generated by the minimal ring of sets containing that ring.

The range of the binaryset function is $\{A,B,\emptyset\}$.

The union of all binary sets is equal to the union of the two sets.

The smallest closed countably determined intersection of sets containing $M$ is contained in the set of all closed countably determined intersections of sets containing $M$.

The smallest closed convex sets containing a set $M$ are a subset of the power set of $\Omega$.

The smallest closed convex decomposition of $\Omega$ is a closed convex decomposition of $\Omega$.

If $M$ is a closed subset of the power set of $\Omega$, then $M$ is a subset of the power set of $\Omega$.

If $\Omega$ is a closed convex set and $M$ is a closed convex cone, then $\Omega - s \in M$ for all $s \in M$.

If $A$ is a chain of sets in a monotone class $\Omega$ and $\Omega$ is closed under countable unions, then $\bigcup A \in \Omega$.

If $\Omega$ is a closed countably decomposable ideal, $M$ is a $\sigma$-algebra, $A$ is a disjoint family of sets in $M$, and the range of $A$ is contained in $M$, then the union of the sets in $A$ is in $M$.

If $M$ is a closed cover of $\Omega$ and $A, B \in M$ are disjoint, then $A \cup B \in M$.

If $A$ and $B$ are disjoint sets in the smallest $\sigma$-algebra containing all the cylinder sets, then $A \cup B$ is also in the smallest $\sigma$-algebra containing all the cylinder sets.