The measure spaces for which all of the nice theorems about $\sigma$-finite measures (except those about product measures, which require other generalizations) generalize are called localizable measure spaces. There are a number of equivalent definitions of localizable measure spaces, perhaps the simplest states that a positive measure $\mu$ is localizable if it is semifinite and has essential suprema, and a complex measure is localizable if its total variation is.

A measure is semifinite if whenever $\mu(E) = \infty$ there exists a measurable $F \subseteq E$ such that $0 < \mu(F) < \infty$. A measure has essential suprema if for every family $\mathcal{E}$ of measurable sets there exists a measurable set $H$ such that $E \setminus H$ is null for every $E \in \mathcal{E}$ and if $G$ is measurable and $E \setminus G$ is negligible for every $E \in \mathcal{E}$, then $H \setminus G$ is negligible.

The most commonly used equivalent definition of a localizable measure space is that real-valued $L^\infty(\mu)$ is Dedekind-complete as a lattice. Essentially all measures that arise in applications are localizable, e.g. all regular Borel measures on locally compact Hausdorff spaces.

The theorem that you would like to generalize is normally called the Lebesgue Decomposition Theorem rather than the Radon-Nikodym Theorem, which is about the representation of absolutely continuous measures in terms of integration against functions. You are missing the hypothesis that $\lambda$ is also $\sigma$-finite, or at least that $\mu + \lambda$ is $\sigma$-finite.

If $\mu$ and $\nu$ are measures on the same measurable space such that $\mu + \nu$ is localizable, then there exists a decomposition $\mu = \mu_c + \mu_s$ where $\mu_c$ is absolutely continuous with respect to $\nu$ and $\mu_s$ is singular. For a simple proof, see Decomposition and Representation Theorems in Measure Theory by Kelley.

The measure spaces for which all of the nice theorems about $\sigma$-finite measures (except those about product measures, which require other generalizations) generalize are called localizable measure spaces. There are a number of equivalent definitions of localizable measure spaces; perhaps the simplest states that a positive measure $\mu$ is localizable if it is semifinite and has essential suprema, and a complex measure is localizable if its total variation is.

A measure is semifinite if whenever $\mu(E) = \infty$ there exists a measurable $F \subseteq E$ such that $0 < \mu(F) < \infty$. A measure has essential suprema if for every family $\mathcal{E}$ of measurable sets there exists a measurable set $H$ such that

• $E \setminus H$ is null for every $E \in \mathcal{E}$, and
• if $G$ is measurable and $E \setminus G$ is negligible for every $E \in \mathcal{E}$, then $H \setminus G$ is negligible.

The most commonly used equivalent definition of a localizable measure space is that real-valued $L^\infty(\mu)$ is Dedekind-complete as a lattice. Essentially all measures that arise in applications are localizable, e.g. all regular Borel measures on locally compact Hausdorff spaces are localizable.

The theorem that you would like to generalize is normally called the Lebesgue Decomposition Theorem rather than the Radon-Nikodym Theorem, which is about the representation of absolutely continuous measures in terms of integration against functions. You are missing the hypothesis that $\lambda$ is also $\sigma$-finite, or at least that $\mu + \lambda$ is $\sigma$-finite.

If $\mu$ and $\nu$ are measures on the same measurable space such that $\mu + \nu$ is localizable, then there exists a decomposition $\mu = \mu_c + \mu_s$ where $\mu_c$ is absolutely continuous with respect to $\nu$ and $\mu_s$ is singular. For a simple proof, see Decomposition and Representation Theorems in Measure Theory by Kelley.

The measure spaces for which all of the nice theorems about $\sigma$-finite measures (except those about product measures, which require other generalizations) generalize are called localizable measure spaces. There are a number of equivalent definitions of localizable measure spaces, perhaps the simplest states that a positive measure $\mu$ is localizable if it is semifinite and has essential suprema, and a complex measure is localizable if its total variation is.

A measure is semifinite if whenever $\mu(E) = \infty$ there exists a measurable $F \subseteq E$ such that $0 < \mu(F) < \infty$. A measure has essential suprema if for every family $\mathcal{E}$ of measurable sets there exists a measurable set $H$ such that $E \setminus H$ is null for every $E \in \mathcal{E}$ and if $G$ is measurable and $E \setminus G$ is negligible for every $E \in \mathcal{E}$, then $H \setminus G$ is negligible.

The most commonly used equivalent definition of a localizable measure space is that real-valued $L^\infty(\mu)$ is Dedekind-complete as a lattice. Essentially all measures that arise in applications are localizable, e.g. all regular Borel measures on locally compact Hausdorff spaces.

The theorem that you would like to generalize is normally called the Lebesgue Decomposition Theorem rather than the Radon-Nikodym Theorem, which is about the representation of absolutely continuous measures in terms of integration against $L^1$ functions. You are missing the hypothesis that $\lambda$ is also $\sigma$-finite, or at least that $\mu + \lambda$ is $\sigma$-finite.

If $\mu$ and $\nu$ are measures on the same measurable space such that $\mu + \nu$ is localizable, then there exists a decomposition $\mu = \mu_c + \mu_s$ where $\mu_c$ is absolutely continuous with respect to $\nu$ and $\mu_s$ is singular. For a simple proof, see Decomposition and Representation Theorems in Measure Theory by Kelley.

The measure spaces for which all of the nice theorems about $\sigma$-finite measures (except those about product measures, which require other generalizations) generalize are called localizable measure spaces. There are a number of equivalent definitions of localizable measure spaces, perhaps the simplest states that a positive measure $\mu$ is localizable if it is semifinite and has essential suprema, and a complex measure is localizable if its total variation is.

A measure is semifinite if whenever $\mu(E) = \infty$ there exists a measurable $F \subseteq E$ such that $0 < \mu(F) < \infty$. A measure has essential suprema if for every family $\mathcal{E}$ of measurable sets there exists a measurable set $H$ such that $E \setminus H$ is null for every $E \in \mathcal{E}$ and if $G$ is measurable and $E \setminus G$ is negligible for every $E \in \mathcal{E}$, then $H \setminus G$ is negligible.

The most commonly used equivalent definition of a localizable measure space is that real-valued $L^\infty(\mu)$ is Dedekind-complete as a lattice. Essentially all measures that arise in applications are localizable, e.g. all regular Borel measures on locally compact Hausdorff spaces.

The theorem that you would like to generalize is normally called the Lebesgue Decomposition Theorem rather than the Radon-Nikodym Theorem, which is about the representation of absolutely continuous measures in terms of integration against functions. You are missing the hypothesis that $\lambda$ is also $\sigma$-finite, or at least that $\mu + \lambda$ is $\sigma$-finite.

If $\mu$ and $\nu$ are measures on the same measurable space such that $\mu + \nu$ is localizable, then there exists a decomposition $\mu = \mu_c + \mu_s$ where $\mu_c$ is absolutely continuous with respect to $\nu$ and $\mu_s$ is singular. For a simple proof, see Decomposition and Representation Theorems in Measure Theory by Kelley.