In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=d\phi|...] \end{align}

Where $\Phi_{k}$ are the atom locations of a random mixing measure, and $\phi$ is some value that $\Phi_{k}$ can take. What is the difference between $\in$ and $=$?

Also, what does the $d\phi$ mean here? I don't really understand the intuition. Particularly, if $G$ is some distribution, what does $G(d\phi)$ mean? I would think it would mean that $G$ is parametrized by whatever is in parentheses, but I don't think that $d\phi$ are parameters.

In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=d\phi|...] \end{align} (see for instance pages 14 and 15 of Peter Orbanz' notes).

Where $\Phi_{k}$ are the atom locations of a random mixing measure, and $\phi$ is some value that $\Phi_{k}$ can take. What is the difference between $\in$ and $=$?

Also, what does the $d\phi$ mean here? I don't really understand the intuition. Particularly, if $G$ is some distribution, what does $G(d\phi)$ mean? I would think it would mean that $G$ is parametrized by whatever is in parentheses, but I don't think that $d\phi$ are parameters.

In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=d\phi|...] \end{align}

Where $\Phi_{k}$ are the atom locations of a random mixing measure, and $\phi$ is some value that $\Phi_{k}$ can take. What is the difference between $\in$ and $\phi$?

Also, what does the $d\phi$ mean here? I don't really understand the intuition. Particularly, if $G$ is some distribution, what does $G(d\phi)$ mean? I would think it would mean that $G$ is parametrized by whatever is in parentheses, but I don't think that $d\phi$ are parameters.

In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=d\phi|...] \end{align}

Where $\Phi_{k}$ are the atom locations of a random mixing measure, and $\phi$ is some value that $\Phi_{k}$ can take. What is the difference between $\in$ and $=$?

Also, what does the $d\phi$ mean here? I don't really understand the intuition. Particularly, if $G$ is some distribution, what does $G(d\phi)$ mean? I would think it would mean that $G$ is parametrized by whatever is in parentheses, but I don't think that $d\phi$ are parameters.

In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=d\phi|...] \end{align}

Where $\Phi_{k}$ are the atom locations of a random mixing measure, and $\phi$ is some value that $\Phi_{k}$ can take. What is the difference between $\in$ and $\phi$?

Also, what does the $d\phi$ mean here? I don't really understand the intuition.

In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=d\phi|...] \end{align}

Where $\Phi_{k}$ are the atom locations of a random mixing measure, and $\phi$ is some value that $\Phi_{k}$ can take. What is the difference between $\in$ and $\phi$?

Also, what does the $d\phi$ mean here? I don't really understand the intuition. Particularly, if $G$ is some distribution, what does $G(d\phi)$ mean? I would think it would mean that $G$ is parametrized by whatever is in parentheses, but I don't think that $d\phi$ are parameters.