It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for Carathéodory) where $\mu_C$ is the restriction of the induced outer measure $\mu^*$ and $\Omega_C$ is the set of all Carathéodory-measurable sets $M$, i.e., $\mu^*(E) = \mu^*(E\cap M) + \mu^*(E\setminus M)$ for all $E\subseteq X$.

Nik Weaver (in Measure Theory and Functional Analysis, Chapter 2, Theorem 2.14) gives an alternate complete extension of the premeasure $(X, \Omega_0, \mu_0)$—let's call it $(X, \Omega_N, \mu_N)$ ($N$ for Nik). I want to compare and hopefully show that the two extensions are the same.

Let me first describe Nik's approach. For him, $\mu_0$ is finite-valued, so let's assume this throughout. He breaks his construction into two cases:

1. $X\in\Omega_0$: Here he declares $\Omega_N$ to be the set of all subsets $M$ of $X$ which can be approximated arbitrarily well by those in $\Omega_0$, i.e., for each $\epsilon > 0$, there exists an $A\in\Omega_0$ such that $\mu^*(M\mathbin{\Delta} A) < \epsilon$. Further, he declares $\mu_N$ to be the restriction of $\mu^*$. Then he shows that $(X, \Omega_N, \mu_N)$ indeed is a complete measure space extending the premeasure.

2. $X\notin\Omega_0$: Here he first defines premeasures $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$ where $\Omega_0^A$ is the set of all those subsets of $\Omega_0$ that are also contained in $A$, and $\mu_0^A$ to simply be the restriction of $\mu_0$. By first case, we then have complete measure spaces $(A, \Omega_N^A, \mu^A_N)$ extending $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$. Finally, he defines \begin{align*} \Omega_N & := \bigcap_{A\in\Omega_0}\{M\subseteq X : M\cap A\in \Omega^A_N\}\text{, and}\\ \mu_N(M) & := \sup_{A\in\Omega_0}\mu_N^A(M\cap N) \end{align*} and shows that $(X, \Omega_N, \mu_N)$ is a complete extension of the original premeasure.

Question: Obviously, one would like to compare Nik's extension to that of Carathéodory. I have successfully shown that the two are equivalent in the first case, namely when $X\in\Omega_0$. However, I have been on the second case for quite some time now, with little progress except showing that $\mu_N\le\mu^*$ on $\Omega_N$. Thus what remains to be shown is the reverse inequality and that $\Omega_N = \Omega_C$. I appreciate any thoughts on this problem.

For the second case, a nice lemma (from Nik's proof) that might come in handy is the following: If $M\in\Omega_N$ is contained in both $A$ and $B$ of $\Omega_0$, then $\mu^A_N(M) = \mu^B_N(M)$. This helps in proving that $\mu_N$ actually extends $\mu_0$.

It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for Carathéodory) where $\mu_C$ is the restriction of the induced outer measure $\mu^*$ and $\Omega_C$ is the set of all Carathéodory-measurable sets $M$, i.e., $\mu^*(E) = \mu^*(E\cap M) + \mu^*(E\setminus M)$ for all $E\subseteq X$.

Nik Weaver (in Measure Theory and Functional Analysis, Chapter 2, Theorem 2.14) gives an alternate complete extension of the premeasure $(X, \Omega_0, \mu_0)$—let's call it $(X, \Omega_N, \mu_N)$ ($N$ for Nik). I want to compare and hopefully show that the two extensions are the same.

Let me first describe Nik's approach. For him, $\mu_0$ is finite-valued, so let's assume this throughout. He breaks his construction into two cases:

1. $X\in\Omega_0$: Here he declares $\Omega_N$ to be the set of all subsets $M$ of $X$ which can be approximated arbitrarily well by those in $\Omega_0$, i.e., for each $\epsilon > 0$, there exists an $A\in\Omega_0$ such that $\mu^*(M\mathbin{\Delta} A) < \epsilon$. Further, he declares $\mu_N$ to be the restriction of $\mu^*$. Then he shows that $(X, \Omega_N, \mu_N)$ indeed is a complete measure space extending the premeasure.

2. $X\notin\Omega_0$: Here he first defines premeasures $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$ where $\Omega_0^A$ is the set of all those subsets of $\Omega_0$ that are also contained in $A$, and $\mu_0^A$ to simply be the restriction of $\mu_0$. By first case, we then have complete measure spaces $(A, \Omega_N^A, \mu^A_N)$ extending $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$. Finally, he defines \begin{align*} \Omega_N & := \bigcap_{A\in\Omega_0}\{M\subseteq X : M\cap A\in \Omega^A_N\}\text{, and}\\ \mu_N(M) & := \sup_{A\in\Omega_0}\mu_N^A(M\cap A) \end{align*} and shows that $(X, \Omega_N, \mu_N)$ is a complete extension of the original premeasure.

Question: Obviously, one would like to compare Nik's extension to that of Carathéodory. I have successfully shown that the two are equivalent in the first case, namely when $X\in\Omega_0$. However, I have been on the second case for quite some time now, with little progress except showing that $\mu_N\le\mu^*$ on $\Omega_N$. Thus what remains to be shown is the reverse inequality and that $\Omega_N = \Omega_C$. I appreciate any thoughts on this problem.

For the second case, a nice lemma (from Nik's proof) that might come in handy is the following: If $M\in\Omega_N$ is contained in both $A$ and $B$ of $\Omega_0$, then $\mu^A_N(M) = \mu^B_N(M)$. This helps in proving that $\mu_N$ actually extends $\mu_0$.

It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for Carathéodory) where $\mu_C$ is the restriction of the induced outer measure $\mu^*$ and $\Omega_C$ is the set of all Carathéodory-measurable sets $M$, i.e., $\mu^*(E) = \mu^*(E\cap M) + \mu^*(E\setminus M)$ for all $E\subseteq X$.

Nik Weaver (in Measure Theory and Functional Analysis, Chapter 2, Theorem 2.14) gives an alternate complete extension of the premeasure $(X, \Omega_0, \mu_0)$—let's call it $(X, \Omega_N, \mu_N)$ ($N$ for Nik). I want to compare and hopefully show that the two extensions are the same.

Let me first describe Nik's approach. For him, $\mu_0$ is finite-valued, so let's assume this throughout. He breaks his construction into two cases:

1. $X\in\Omega_0$: Here he declares $\Omega_N$ to be the set of all subsets $M$ of $X$ which can be approximated arbitrarily well by those in $\Omega_0$, i.e., for each $\epsilon > 0$, there exists an $A\in\Omega_0$ such that $\mu^*(M\mathbin{\Delta} A) < \epsilon$. Further, he declares $\mu_N$ to be the restriction of $\mu^*$. Then he shows that $(X, \Omega_N, \mu_N)$ indeed is a complete measure space extending the premesaure.

2. $X\notin\Omega_0$: Here he first defines premeasures $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$ where $\Omega_0^A$ is the set of all those subsets of $\Omega_0$ that are also contained in $A$, and $\mu_0^A$ to simply be the restriction of $\mu_0$. By first case, we then have complete measure spaces $(A, \Omega_N^A, \mu^A_N)$ extending $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$. Finally, he defines \begin{align*} \Omega_N & := \bigcap_{A\in\Omega_0}\{M\subseteq X : M\cap A\in \Omega^A_N\}\text{, and}\\ \mu_N(M) & := \sup_{A\in\Omega_0}\mu_N^A(M\cap N) \end{align*} and shows that $(X, \Omega_N, \mu_N)$ is a complete extension of the original premeasure.

Question: Obviously, one would like to compare Nik's extension to that of Carathéodory. I have successfully shown that the two are equivalent in the first case, namely when $X\in\Omega_0$. However, I have been on the second case for quite some time now, with little progress except showing that $\mu_N\le\mu^*$ on $\Omega_N$. Thus what remains to be shown is the reverse inequality and that $\Omega_N = \Omega_C$. I appreciate any thoughts on this problem.

For the second case, a nice lemma (from Nik's proof) that might come in handy is the following: If $M\in\Omega_N$ is contained in both $A$ and $B$ of $\Omega_0$, then $\mu^A_N(M) = \mu^B_N(M)$. This helps in proving that $\mu_N$ actually extends $\mu_0$.

It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for Carathéodory) where $\mu_C$ is the restriction of the induced outer measure $\mu^*$ and $\Omega_C$ is the set of all Carathéodory-measurable sets $M$, i.e., $\mu^*(E) = \mu^*(E\cap M) + \mu^*(E\setminus M)$ for all $E\subseteq X$.

Nik Weaver (in Measure Theory and Functional Analysis, Chapter 2, Theorem 2.14) gives an alternate complete extension of the premeasure $(X, \Omega_0, \mu_0)$—let's call it $(X, \Omega_N, \mu_N)$ ($N$ for Nik). I want to compare and hopefully show that the two extensions are the same.

Let me first describe Nik's approach. For him, $\mu_0$ is finite-valued, so let's assume this throughout. He breaks his construction into two cases:

1. $X\in\Omega_0$: Here he declares $\Omega_N$ to be the set of all subsets $M$ of $X$ which can be approximated arbitrarily well by those in $\Omega_0$, i.e., for each $\epsilon > 0$, there exists an $A\in\Omega_0$ such that $\mu^*(M\mathbin{\Delta} A) < \epsilon$. Further, he declares $\mu_N$ to be the restriction of $\mu^*$. Then he shows that $(X, \Omega_N, \mu_N)$ indeed is a complete measure space extending the premeasure.

2. $X\notin\Omega_0$: Here he first defines premeasures $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$ where $\Omega_0^A$ is the set of all those subsets of $\Omega_0$ that are also contained in $A$, and $\mu_0^A$ to simply be the restriction of $\mu_0$. By first case, we then have complete measure spaces $(A, \Omega_N^A, \mu^A_N)$ extending $(A, \Omega_0^A, \mu_0^A)$ for each $A\in\Omega_0$. Finally, he defines \begin{align*} \Omega_N & := \bigcap_{A\in\Omega_0}\{M\subseteq X : M\cap A\in \Omega^A_N\}\text{, and}\\ \mu_N(M) & := \sup_{A\in\Omega_0}\mu_N^A(M\cap N) \end{align*} and shows that $(X, \Omega_N, \mu_N)$ is a complete extension of the original premeasure.

Question: Obviously, one would like to compare Nik's extension to that of Carathéodory. I have successfully shown that the two are equivalent in the first case, namely when $X\in\Omega_0$. However, I have been on the second case for quite some time now, with little progress except showing that $\mu_N\le\mu^*$ on $\Omega_N$. Thus what remains to be shown is the reverse inequality and that $\Omega_N = \Omega_C$. I appreciate any thoughts on this problem.

For the second case, a nice lemma (from Nik's proof) that might come in handy is the following: If $M\in\Omega_N$ is contained in both $A$ and $B$ of $\Omega_0$, then $\mu^A_N(M) = \mu^B_N(M)$. This helps in proving that $\mu_N$ actually extends $\mu_0$.