- #1

- 141

- 5

**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.

Last edited: