Sigma-algebra used in the theorem of Lebesgue-Radon-Nikodym of Rudin's Real and Complex Analysis

Question

The theorem of Lebegue-Radon-Nikodym in page 121 of Rudin's Real and Complex Analysis reads as:

Let $\mu$ be a positive $\sigma$-finite measure on a $\sigma$-algebra $\mathfrak{M}$ on a set $X$, and let $\lambda$ be the complex measure on $\mathfrak{M}$.

1. Then there is a unique pair of complex measures $\lambda_a,\lambda_s$ on $\mathfrak{M}$, called the \textbf{Lebesgue decomposition} of $\lambda$ relative to $\mu$, such that

• $\lambda = \lambda_a + \lambda_s$
• $\lambda_a\ll \mu$
• $\lambda_s\perp \mu$

2. There is a unique $h\in L^1(\mu)$ such that $$\lambda_a(E) = \int_Ehd\mu$$ for every $E\in\mathfrak{M}$.

Rudin seems to be keen on using the symbol $\mathfrak{M}$ as a sigma-algebra throughout RCA. At first I thought that in the theorem of Lebesgue-Radon-Nikodym, $\mathfrak{M}$ is any sigma-algebra. However, prior to LRN, Rudin has introduced the Riesz Representation Theorem on page 40, in which

1. $X$ is a locally compact Hausdorff space

2. $\mathfrak{M}_F$ as the class of all $E\subset X$ which satisfy i.) $\mu(E) < \infty$ and ii.) $\mu(E) = \sup\{\mu(K):K\subset E,\text{$K$ compact}\}$

3. $\mathfrak{M}$ is the class of all $E\subset X$ such that $E\cap K\in \mathfrak{M}_F$ for every compact set $K$

(Question:) Is the sigma-algebra $\mathfrak{M}$ defined in the Riesz Representation Theorem related in any way to the sigma-algebra used in the theorem of Lebesgue-Radon-Nikodym? Or can one use really any sigma-algebra in LRN?

Answer 1

No relationship, and any $\sigma$-algebra can be used for the Lebesgue Decomposition and Radon-Nikodym derivative.

Note also that the Riesz Representation Theorem requires $X$ to be a topological space, which is not a requirement in Lebesgue-Radon-Nikodym, so the old $\mathfrak M$ makes no sense in the new theorem.