There are surely many directions that you could take this thought, but here is one of them. Note that the formula you have given for the joint distribution of a sample of size $\beta$ with a continuous empirical distribution closely resembles the definition the relative entropy of a distribution $\rho$ with respect to another distribution $\mu$: $$H(\rho | \mu) = \int\log(d\rho/d\mu)d\rho. $$ In fact, for any sample $\{x_i\}$ of size $n$ if we let $p$ denote the empirical distribution $\frac{1}{n}\sum \delta(x_i)$, then the log joint density is $$\sum \log(d\mu(x_i))=n\int \log(d\mu)dp.$$ This formula gives you joint probability density at a given point in $\mathbb R^n$, but it doesn't accurately measure the "probability" of the resulting empirical distribution, because it fails to take into account the possible permutations of the sample points. Using Stirling's approximation, and letting $n_i$ denote the number of sample points at each distinct value, we may asymptotically approximate the number of permutations which yield the same empirical distribution $$\log\frac{n!}{n_1!n_2!\cdots n_k!}\simeq n\log n - \sum n_i\log n_i = -n \left(\sum p_i \log p_i\right). $$

Putting these two formulas together yields an estimate of the "probability" of generating a specific empirical distribution (with considerable abuse): $$n\int \left(\log(d\mu)-\log(p)\right)dp. $$ Note that we can't write this in terms of relative entropy as $-nH(p| \mu)$, because $p$ is not absolutely continuous with respect to $\mu$, so the Radon-Nikodym derivative is not well defined. However, if the empirical distribution $p$ were in fact a continuous distribution, as you have in your research problem, then this formula would begin to make some sense.

These ideas have in fact been made rigorous in the theory of large deviations. Let $\mathcal P (\mathbb R)$ denote the set of all probability distributions on $\mathbb R$, under the topology of weak convergence. Let $\mu\in \mathcal P(\mathbb R)$ be any probability distribution, and let $\mu_n$ be a random element in $\mathcal P(\mathbb R)$ corresponding to the empirical distribution on an i.i.d sample of size $n$ drawn according to $\mu$.

As $n\rightarrow\infty$, $\mu_n$ will converge in probability to $\mu$, and the rate of this convergence can be described by a large deviations principle, for which the rate function is the relative entropy with respect to $\mu$. What this means is that, for any measurable set $B\in \mathcal P(\mathbb R)$,

$$\begin{align} \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\geq -\inf_{\rho\in B^o}H(\rho|\mu)\\ \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\leq -\inf_{\rho\in \bar{B}}H(\rho|\mu), \end{align}$$ where $B^o$ and $\bar{B}$ are the interior and closure of $B$.

So this formalizes the idea that the probability of drawing a sample of size $n$ from $\mu$ and ending up with an empirical distribution in the neighborhood of $\rho$ is approximately $e^{-nH(\rho|\mu)}$.

As far as what a "fractional sample" actually means, I'm not sure it matters any more than what the "factorial" of a non-integer is, or what the "square root" of a negative number is, provided that it is a useful mathematical construct.

There are surely many directions that you could take this thought, but here is one of them. Note that the formula you have given for the joint distribution of a sample of size $\beta$ with a continuous empirical distribution closely resembles the definition the relative entropy of a distribution $\rho$ with respect to another distribution $\mu$: $$H(\rho | \mu) = \int\log(d\rho/d\mu)d\rho. $$ In fact, for any sample $\{x_i\}$ of size $n$ if we let $p$ denote the empirical distribution $\frac{1}{n}\sum \delta(x_i)$, then the log joint density is $$\sum \log(d\mu(x_i))=n\int \log(d\mu)dp.$$ This formula gives you joint probability density at a given point in $\mathbb R^n$, but it doesn't accurately measure the "probability" of the resulting empirical distribution, because it fails to take into account the possible permutations of the sample points. Using Stirling's approximation, and letting $n_i$ denote the number of sample points at each distinct value, we may asymptotically approximate the number of permutations which yield the same empirical distribution $$\log\frac{n!}{n_1!n_2!\cdots n_k!}\simeq n\log n - \sum n_i\log n_i = -n \left(\sum p_i \log p_i\right). $$

Putting these two formulas together yields an estimate of the "probability" of generating a specific empirical distribution (with considerable abuse of notation): $$n\int \left(\log(d\mu)-\log(p)\right)dp. $$ Note that we can't write this in terms of relative entropy as $-nH(p| \mu)$, because $p$ is not absolutely continuous with respect to $\mu$, so the Radon-Nikodym derivative is not well defined. However, if the empirical distribution $p$ were in fact a continuous distribution, as you have in your research problem, then this formula would begin to make some sense.

These ideas have in fact been made rigorous in the theory of large deviations. Let $\mathcal P (\mathbb R)$ denote the set of all probability distributions on $\mathbb R$, under the topology of weak convergence. Let $\mu\in \mathcal P(\mathbb R)$ be any probability distribution, and let $\mu_n$ be a random element in $\mathcal P(\mathbb R)$ corresponding to the empirical distribution on an i.i.d sample of size $n$ drawn according to $\mu$.

As $n\rightarrow\infty$, $\mu_n$ will converge in probability to $\mu$, and the rate of this convergence can be described by a large deviations principle, for which the rate function is the relative entropy with respect to $\mu$. What this means is that, for any measurable set $B\in \mathcal P(\mathbb R)$,

$$\begin{align} \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\geq -\inf_{\rho\in B^o}H(\rho|\mu)\\ \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\leq -\inf_{\rho\in \bar{B}}H(\rho|\mu), \end{align}$$ where $B^o$ and $\bar{B}$ are the interior and closure of $B$.

So this formalizes the idea that the probability of drawing a sample of size $n$ from $\mu$ and ending up with an empirical distribution in the neighborhood of $\rho$ is approximately $e^{-nH(\rho|\mu)}$.

As far as what a "fractional sample" actually means, I'm not sure it matters any more than what the "factorial" of a non-integer is, or what the "square root" of a negative number is, provided that it is a useful mathematical construct.

There are surely many directions that you could take this thought, but here is one of them. Note that the formula you have given for the joint distribution of a sample of size $\beta$ with a continuous empirical distribution closely resembles the definition the relative entropy of a distribution $\rho$ with respect to another distribution $\mu$: $$H(\rho | \mu) = \int\log(d\rho/d\mu)d\rho. $$ In fact, for any sample $\{x_i\}$ of size $n$ if we let $p$ denote the empirical distribution $\frac{1}{n}\sum \delta(x_i)$, then the log joint density is $$\sum \log(d\mu(x_i))=n\int \log(d\mu)dp.$$ This formula gives you joint probability density at a given point in $\mathbb R^n$, but it doesn't accurately measure the "probability" of the resulting empirical distribution, because it fails to take into account the possible permutations of the sample points. Using Stirling's approximation, and letting $n_i$ denote the number of sample points at each distinct value, we may asymptotically approximate the number of permutations which yield the same empirical distribution $$\log\frac{n!}{n_1!n_2!\cdots n_k!}\simeq n\log n - \sum n_i\log n_i = -n \left(\sum p_i \log p_i\right). $$

Putting these two formulas together yields an estimate of the "probability" of generating a specific empirical distribution (with considerable abuse): $$n\int \left(\log(d\mu)-\log(dp)\right)dp. $$ Note that we can't write this in terms of relative entropy as $-nH(p| \mu)$, because $p$ is not absolutely continuous with respect to $\mu$, so the Radon-Nikodym derivative is not well defined. However, if the empirical distribution $p$ were in fact a continuous distribution, as you have in your research problem, then this formula would begin to make some sense.

These ideas have in fact been made rigorous in the theory of large deviations. Let $\mathcal P (\mathbb R)$ denote the set of all probability distributions on $\mathbb R$, under the topology of weak convergence. Let $\mu\in \mathcal P(\mathbb R)$ be any probability distribution, and let $\mu_n$ be a random element in $\mathcal P(\mathbb R)$ corresponding to the empirical distribution on an i.i.d sample of size $n$ drawn according to $\mu$.

As $n\rightarrow\infty$, $\mu_n$ will converge in probability to $\mu$, and the rate of this convergence can be described by a large deviations principle, for which the rate function is the relative entropy with respect to $\mu$. What this means is that, for any measurable set $B\in \mathcal P(\mathbb R)$,

$$\begin{align} \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\geq -\inf_{\rho\in B^o}H(\rho|\mu)\\ \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\leq -\inf_{\rho\in \bar{B}}H(\rho|\mu), \end{align}$$ where $B^o$ and $\bar{B}$ are the interior and closure of $B$.

So this formalizes the idea that the probability of drawing a sample of size $n$ from $\mu$ and ending up with an empirical distribution in the neighborhood of $\rho$ is approximately $e^{-nH(\rho|\mu)}$.

As far as what a "fractional sample" actually means, I'm not sure it matters any more than what the "factorial" of a non-integer is, or what the "square root" of a negative number is, provided that it is a useful mathematical construct.

There are surely many directions that you could take this thought, but here is one of them. Note that the formula you have given for the joint distribution of a sample of size $\beta$ with a continuous empirical distribution closely resembles the definition the relative entropy of a distribution $\rho$ with respect to another distribution $\mu$: $$H(\rho | \mu) = \int\log(d\rho/d\mu)d\rho. $$ In fact, for any sample $\{x_i\}$ of size $n$ if we let $p$ denote the empirical distribution $\frac{1}{n}\sum \delta(x_i)$, then the log joint density is $$\sum \log(d\mu(x_i))=n\int \log(d\mu)dp.$$ This formula gives you joint probability density at a given point in $\mathbb R^n$, but it doesn't accurately measure the "probability" of the resulting empirical distribution, because it fails to take into account the possible permutations of the sample points. Using Stirling's approximation, and letting $n_i$ denote the number of sample points at each distinct value, we may asymptotically approximate the number of permutations which yield the same empirical distribution $$\log\frac{n!}{n_1!n_2!\cdots n_k!}\simeq n\log n - \sum n_i\log n_i = -n \left(\sum p_i \log p_i\right). $$

Putting these two formulas together yields an estimate of the "probability" of generating a specific empirical distribution (with considerable abuse): $$n\int \left(\log(d\mu)-\log(p)\right)dp. $$ Note that we can't write this in terms of relative entropy as $-nH(p| \mu)$, because $p$ is not absolutely continuous with respect to $\mu$, so the Radon-Nikodym derivative is not well defined. However, if the empirical distribution $p$ were in fact a continuous distribution, as you have in your research problem, then this formula would begin to make some sense.

These ideas have in fact been made rigorous in the theory of large deviations. Let $\mathcal P (\mathbb R)$ denote the set of all probability distributions on $\mathbb R$, under the topology of weak convergence. Let $\mu\in \mathcal P(\mathbb R)$ be any probability distribution, and let $\mu_n$ be a random element in $\mathcal P(\mathbb R)$ corresponding to the empirical distribution on an i.i.d sample of size $n$ drawn according to $\mu$.

As $n\rightarrow\infty$, $\mu_n$ will converge in probability to $\mu$, and the rate of this convergence can be described by a large deviations principle, for which the rate function is the relative entropy with respect to $\mu$. What this means is that, for any measurable set $B\in \mathcal P(\mathbb R)$,

$$\begin{align} \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\geq -\inf_{\rho\in B^o}H(\rho|\mu)\\ \liminf_{n\rightarrow \infty}\frac{1}{n} \log P[\mu_n\in B]&\leq -\inf_{\rho\in \bar{B}}H(\rho|\mu), \end{align}$$ where $B^o$ and $\bar{B}$ are the interior and closure of $B$.

So this formalizes the idea that the probability of drawing a sample of size $n$ from $\mu$ and ending up with an empirical distribution in the neighborhood of $\rho$ is approximately $e^{-nH(\rho|\mu)}$.

As far as what a "fractional sample" actually means, I'm not sure it matters any more than what the "factorial" of a non-integer is, or what the "square root" of a negative number is, provided that it is a useful mathematical construct.