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SqueeSpleen
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Measure Theory, Caratheodory condition
The set [itex]E \subset ℝ^{p}[/itex] satisfy Caratheodory's condition if:
[itex]\forall A \subset ℝ^{p}[/itex]
[itex]m_e (A) = m_e(A \cap E) + m_e(A \cap E^c)[/itex]
Prove that if E is measurable then E satisfy the Caratheodory's condition.
I know
[itex]m_e (A) \leq m_e(A \cap E) + m_e(A \cap E^c)[/itex] for the subaditivity of the external measure, but I didn't found a way to prove:
[itex]m_e (A) \geq m_e(A \cap E) + m_e(A \cap E^c)[/itex]
I'll post again if I figure out something, but I already skipped this some days ago because I was an entire hour without any advance at all.
Feel free to make any correction's, it's the first time I write something about this subject in English and I'm not sure if I'm chosing the correct words.
The set [itex]E \subset ℝ^{p}[/itex] satisfy Caratheodory's condition if:
[itex]\forall A \subset ℝ^{p}[/itex]
[itex]m_e (A) = m_e(A \cap E) + m_e(A \cap E^c)[/itex]
Prove that if E is measurable then E satisfy the Caratheodory's condition.
I know
[itex]m_e (A) \leq m_e(A \cap E) + m_e(A \cap E^c)[/itex] for the subaditivity of the external measure, but I didn't found a way to prove:
[itex]m_e (A) \geq m_e(A \cap E) + m_e(A \cap E^c)[/itex]
I'll post again if I figure out something, but I already skipped this some days ago because I was an entire hour without any advance at all.
Feel free to make any correction's, it's the first time I write something about this subject in English and I'm not sure if I'm chosing the correct words.
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