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Smallest Sigma Algebra .... Axler, Example 2.28 ....
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[QUOTE="Math Amateur, post: 6375830, member: 203675"] Given that ##X## is a set and ##\mathcal{A} = \{ \{ x \} : x \in X \}## ... ... Require to demonstrate that ##\mathcal{S}:=\{E \subseteq X: |E| < \infty \lor |E^c| < \infty\}## is a ##\sigma##-algebra containing ##\mathcal{A}##[B][U]Proof:[/U][/B] First show that ##\mathcal{S}## contains ##\mathcal{A}## ... Now ##E \subseteq \mathcal{A} \Longrightarrow E = \{ x \}## for some ##x \in X## ... ... ##\Longrightarrow \mid E \mid = 0 \lt \infty## ... ##\Longrightarrow E \subseteq \mathcal{S}## Now ... show ##\mathcal{S}## is a ##\sigma##-algebra ... ##\bullet## Require ##\emptyset \in \mathcal{S}## ... ... suppose ##E = \emptyset## ... ... then ##\mid E \mid = 0 \lt \infty## ... ... so ##\emptyset \in \mathcal{S}## ...##\bullet## Require that if ##E \in \mathcal{S}## then ##E^c \in \mathcal{S}## ... Suppose ##E \in \mathcal{S}## ... Then ##E^c \in \mathcal{S}## ... since ##\mid E^c \mid = 0 \lt \infty## or ##\mid (E^c)^c \mid = \mid E \mid \lt \infty## ... one of which is true since ##E \in \mathcal{S}## ... ... so ##E \in \mathcal{S}## then ##E^c \in \mathcal{S}## ... ##\bullet## Require that if ##E_1, E_2##, ... ... is a sequence of elements of ##\mathcal{S}## ... then ##\bigcup_{ k = 1 }^{ \infty } \in \mathcal{S}## ... Assume that ##E_1, E_2##, ... ... is a sequence of elements of ##\mathcal{S}## ... Then ##\mid E_1 \mid = n_1 \lt \infty## ... or ... ##\mid E_1^c \mid = m_1 \lt \infty## ... ... ... and ##\mid E_2 \mid = n_2 \lt \infty## ... or ... ##\mid E_2^c \mid = m_2 \lt \infty## ... ... .. and so on ... ... But how to proceed from here ...? Can you help further ...? Peter =========================================================================== My thoughts on proceeding ...We could divide ##E_1, E_2##, ... ... into two subsequences ##E_{k_1}, E_{k_2}##, ... and ##E_{h_1}, E_{h_2}, ...## where the subsequence ##E_{k_1}, E_{k_2}##, ... is composed of all those elements of ##\mathcal{S}## ... where ##\mid E_k \mid \lt \infty## ... and where the subsequence ##E_{h_1}, E_{h_2}##, ... is composed of all those elements of ##\mathcal{S}## ... where ##\mid E_k^c \mid \lt \infty## ... and try to show using countable additivity that the unions of the elements of both subsequences are less than infinity and try to relate these results to the union of the elements of E_1, E_2, ... ... [/QUOTE]
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Smallest Sigma Algebra .... Axler, Example 2.28 ....
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