# Proving Caratheodory's Condition for Measurable Sets in Measure Theory

• SqueeSpleen
In summary, the Caratheodory condition states that a set is measurable if its elements satisfy a certain condition.

#### SqueeSpleen

Measure Theory, Caratheodory condition

The set $E \subset ℝ^{p}$ satisfy Caratheodory's condition if:
$\forall A \subset ℝ^{p}$
$m_e (A) = m_e(A \cap E) + m_e(A \cap E^c)$
Prove that if E is measurable then E satisfy the Caratheodory's condition.
I know
$m_e (A) \leq m_e(A \cap E) + m_e(A \cap E^c)$ for the subaditivity of the external measure, but I didn't found a way to prove:
$m_e (A) \geq m_e(A \cap E) + m_e(A \cap E^c)$

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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SqueeSpleen said:
The set $E \subset ℝ^{p}$ satisfy Caratheodory's condition if:
$\forall A \subset ℝ^{p}$
$m_e (A) = m_e(A \cap E) + m_e(A \cap E^c)$
Prove that if E is measurable then E satisfy the Caratheodory's condition.
I know
$m_e (A) \leq m_e(A \cap E) + m_e(A \cap E^c)$ for the subaditivity of the external measure, but I didn't found a way to prove:
$m_e (A) \geq m_e(A \cap E) + m_e(A \cap E^c)$

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.

Some books present the Caratheodory condition as the definition of measurability of E. What definition does your book use?

A set is measurable if $\forall \epsilon > 0$ $\exists U \in \cal{U}$ such that:
$A \subset U$ and $m_e (U-A) < \epsilon$
$\cal{U}$ is the family of the $\sigma$-elemental sets.
I also know that a set E is measurable then $m_i(E)=m_e(E)$

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On a related topic (I don't know if I'd better create a new thread or not).

I have to find a decreasing sequense $\left \{ E_k \right \}_{k \in \mathbb{N}} \subset P(R^p)$ such that $m_e(\displaystyle\bigcap\limits_{i=0}^{\infty} E_k) < \lim_{k \rightarrow \infty}m_e(E_k)$
I think that the following sequence would work:
Notation: $\overline{B}(c,r)=\left [ c-r,c+r \right ]$
$E_1 = \overline{B}(0,1)$
$E_2 = \overline{B}(0,\frac{1}{2}) \cup \overline{B}(1,\frac{1}{2})$
...
$E_n = \displaystyle\bigcup\limits_{i=0}^{n-1} \overline{B}(\frac{k}{n},\frac{1}{n})$
What I'm trying is to get a set which intersection is countable then of null measure.
While the limit of its measures is 1.
It's this idea right or I'd better to search other way of solving this? (If feel like it's not, I think that the intersection will not be countable, it will be [0,1] but I liked this idea so I needed to write it clearly to see that's not good).
I'd try to do it with a no measurable set?
PD: The correct term is "outer measure" not "exterior measure"? Or both are right?

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## 1. What is the Caratheodory condition?

The Caratheodory condition is a mathematical concept used in the study of thermodynamics and statistical mechanics. It states that for a system to be in thermodynamic equilibrium, its entropy must be a differentiable function of its internal energy and volume.

## 2. Who is the scientist behind the Caratheodory condition?

The Caratheodory condition is named after the Greek mathematician Constantin Caratheodory.

## 3. How is the Caratheodory condition used in thermodynamics?

In thermodynamics, the Caratheodory condition is used to define the thermodynamic state variables of a system, such as entropy, internal energy, and volume. It also helps to determine the conditions for a system to be in equilibrium.

## 4. Can the Caratheodory condition be applied to all systems?

No, the Caratheodory condition is only applicable to systems that are in thermodynamic equilibrium. It cannot be applied to systems that are out of equilibrium or undergoing a phase transition.

## 5. What are some real-world examples of the Caratheodory condition?

The Caratheodory condition can be observed in various physical systems, such as a gas in a container, a liquid in a closed bottle, or a solid in a sealed box. It also applies to more complex systems, such as the Earth's atmosphere and the human body.