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Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.

To clarify, by "isomorphic to a measurable subset", I mean that there should exist a measurable injection $X\hookrightarrow\{0,1\}^\kappa$ which in addition takes measurable sets to measurable sets. As pointed out in the comments, there are two $\sigma$-algebras which one can put on $\{0,1\}^\kappa$, and these two differ for uncountable $\kappa$:

1. The Baire $\sigma$-algebra, which is the smallest $\sigma$-algebra making the product projections $\{0,1\}^\kappa\to\{0,1\}$ measurable;
2. The Borel $\sigma$-algebra, which is the one generated by the product topology.

These two give rise to different versions of my question, and it would be optimal to have an answer in both cases, although I currently find the Baire $\sigma$-algebra more natural.

For example, if $(X,\Sigma)$ is the standard Borel space, then either kind of embedding exists since $\{0,1\}^{\mathbb{N}}$ is itself a standard Borel space.

Clearly, a necessary condition for either embedding to exist is that $(X,\Sigma)$ needs to be separated in the sense that for all $x,y\in X$, there is $A\in\Sigma$ with $x\in A$ and $y\not\in A$.

Is separatedness sufficient for $(X,\Sigma)$ to embed into some $\{0,1\}^\kappa$, equipped with either the Baire or the Borel $\sigma$-algebra?

Motivation: I am working with Bayesian networks containing latent variables. I want to allow the sample space of a latent variable to be an arbitrary probability space. Since the latter are difficult to work with, I am trying to make an argument along the lines of replacing every latent variable by a collection of binary variables. A positive answer to the above question would be one way to do this.

Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.

To clarify, by "isomorphic to a measurable subset", I mean that there should exist a measurable injection $X\hookrightarrow\{0,1\}^\kappa$ which in addition takes measurable sets to measurable sets. As pointed out in the comments, there are two $\sigma$-algebras which one can put on $\{0,1\}^\kappa$, and these two differ for uncountable $\kappa$:

1. The Baire $\sigma$-algebra, which is the smallest $\sigma$-algebra making the product projections $\{0,1\}^\kappa\to\{0,1\}$ measurable;
2. The Borel $\sigma$-algebra, which is the one generated by the product topology.

These two give rise to different versions of my question, and it would be optimal to have an answer in both cases, although I currently find the Baire $\sigma$-algebra more natural.

For example, if $(X,\Sigma)$ is the standard Borel space, then either kind of embedding exists since $\{0,1\}^{\mathbb{N}}$ is itself a standard Borel space.

Clearly, a necessary condition for either embedding to exist is that $(X,\Sigma)$ needs to be separated in the sense that for all $x,y\in X$, there is $A\in\Sigma$ with $x\in A$ and $y\not\in A$.

Is separatedness sufficient for $(X,\Sigma)$ to embed into some $\{0,1\}^\kappa$, equipped with either the Baire or the Borel $\sigma$-algebra?

Motivation: I am working with Bayesian networks containing latent variables. I want to allow the sample space of a latent variable to be an arbitrary probability space. Since the latter are difficult to work with, I am trying to make an argument along the lines of replacing every latent variable by a collection of binary variables. A positive answer to the above question would be one way to do this.

Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.

To clarify, $\{0,1\}^\kappa$ is equipped with the usual $\sigma$-algebra for infinite products, which is the smallest $\sigma$-algebra making all component projections $\{0,1\}^\kappa\to\{0,1\}$ measurable. And by "isomorphic to a measurable subset", I mean that there should exist a measurable injection $X\hookrightarrow\{0,1\}^\kappa$ which also takes measurable sets to measurable sets.

For example, such an embedding trivially exists for a standard Borel space, since $\{0,1\}^{\mathbb{N}}$ is itself a standard Borel space.

Clearly, a necessary condition is that $(X,\Sigma)$ needs to be separated in the sense that for all $x,y\in X$, there is $A\in\Sigma$ with $x\in A$ and $y\not\in A$.

Is separatedness sufficient for $(X,\Sigma)$ to embed into some $\{0,1\}^\kappa$?

There always is an obvious measurable map $X\hookrightarrow\{0,1\}^\Sigma$, but I don't even see why it should have measurable image, and I suspect that in general it doesn't. Hence I am at a loss as to how to approach this question. Thanks!

Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.

To clarify, by "isomorphic to a measurable subset", I mean that there should exist a measurable injection $X\hookrightarrow\{0,1\}^\kappa$ which in addition takes measurable sets to measurable sets. As pointed out in the comments, there are two $\sigma$-algebras which one can put on $\{0,1\}^\kappa$, and these two differ for uncountable $\kappa$:

1. The Baire $\sigma$-algebra, which is the smallest $\sigma$-algebra making the product projections $\{0,1\}^\kappa\to\{0,1\}$ measurable;
2. The Borel $\sigma$-algebra, which is the one generated by the product topology.

These two give rise to different versions of my question, and it would be optimal to have an answer in both cases, although I currently find the Baire $\sigma$-algebra more natural.

For example, if $(X,\Sigma)$ is the standard Borel space, then either kind of embedding exists since $\{0,1\}^{\mathbb{N}}$ is itself a standard Borel space.

Clearly, a necessary condition for either embedding to exist is that $(X,\Sigma)$ needs to be separated in the sense that for all $x,y\in X$, there is $A\in\Sigma$ with $x\in A$ and $y\not\in A$.

Is separatedness sufficient for $(X,\Sigma)$ to embed into some $\{0,1\}^\kappa$, equipped with either the Baire or the Borel $\sigma$-algebra?

Motivation: I am working with Bayesian networks containing latent variables. I want to allow the sample space of a latent variable to be an arbitrary probability space. Since the latter are difficult to work with, I am trying to make an argument along the lines of replacing every latent variable by a collection of binary variables. A positive answer to the above question would be one way to do this.

Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.

For example, such an embedding trivially exists for a standard Borel space, since $\{0,1\}^{\mathbb{N}}$ is itself a standard Borel space.

Clearly, a necessary condition is that $(X,\Sigma)$ needs to be separated in the sense that for all $x,y\in X$, there is $A\in\Sigma$ with $x\in A$ and $y\not\in A$.

Is separatedness sufficient for $(X,\Sigma)$ to embed into some $\{0,1\}^\kappa$?

There is an obvious measurable map $X\hookrightarrow\{0,1\}^\Sigma$, but I don't see why it should have measurable image, and I suspect that in general it doesn't. Hence I am at a loss as to how to approach this question. Thanks!

Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.

To clarify, $\{0,1\}^\kappa$ is equipped with the usual $\sigma$-algebra for infinite products, which is the smallest $\sigma$-algebra making all component projections $\{0,1\}^\kappa\to\{0,1\}$ measurable. And by "isomorphic to a measurable subset", I mean that there should exist a measurable injection $X\hookrightarrow\{0,1\}^\kappa$ which also takes measurable sets to measurable sets.

For example, such an embedding trivially exists for a standard Borel space, since $\{0,1\}^{\mathbb{N}}$ is itself a standard Borel space.

Clearly, a necessary condition is that $(X,\Sigma)$ needs to be separated in the sense that for all $x,y\in X$, there is $A\in\Sigma$ with $x\in A$ and $y\not\in A$.

Is separatedness sufficient for $(X,\Sigma)$ to embed into some $\{0,1\}^\kappa$?

There always is an obvious measurable map $X\hookrightarrow\{0,1\}^\Sigma$, but I don't even see why it should have measurable image, and I suspect that in general it doesn't. Hence I am at a loss as to how to approach this question. Thanks!