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In the context of asymptotic results involving the maximum likelihood estimator, the use of the subscript $\theta_0$ in the expectations operator means the following,

$$\mathbb{E}_{\theta_0}[l(\theta; X)] = \mathbb{E}_{X \sim f_X(x; \theta_0)}[l(\theta, X)] = \int l(\theta; x)f_X(x; \theta_0) \space dx,$$

where $f_X$ is the probability density of $X$, parametrised by $\theta_0$, and where I've omitted the range of integration.

As a point of emphasis, within this frequentist context, you are entirely correct in identifying that we do not assume that $\theta_0$ is in any way a random variable. Nor at any stage would it be coherent to consequently speak of a density function on $\theta_0$ with respect to which we are computing expectations.

In contrast with the widely used notation $\mathbb{E}_X[g(X)] = \mathbb{E}_{X \sim f_X(x)}[g(X)]$, where $f_X$ is again a density, the notation $\mathbb{E}_{\theta_0}[\cdot]$ can take some getting used to, as is the case with most issues concerning notation.

In the context of asymptotic results involving the maximum likelihood estimator, the use of the subscript $\theta_0$ in the expectations operator means the following,

$$\mathbb{E}_{\theta_0}[l(\theta; X)] = \mathbb{E}_{X \sim f_X(x; \theta_0)}[l(\theta; X)] = \int l(\theta; x)f_X(x; \theta_0) \space dx,$$

where $f_X$ is the probability density of $X$, parametrised by $\theta_0$, and where I've omitted the limits of integration.

As a point of emphasis, within this frequentist context, you are entirely correct in identifying that we do not assume that $\theta_0$ is in any way a random variable. Nor at any stage would it be coherent to consequently speak of a density function on $\theta_0$ with respect to which we are computing expectations.

In contrast with the widely used notation $\mathbb{E}_X[g(X)] = \mathbb{E}_{X \sim f_X(x)}[g(X)]$, where $f_X$ is again a density, the notation $\mathbb{E}_{\theta_0}[\cdot]$ can take some getting used to, as is the case with most issues concerning notation.

In the context of asymptotic results involving the maximum likelihood estimator, the use of the subscript $\theta_0$ in the expectations operator means the following,

$$\mathbb{E}_{\theta_0}[l(\theta; X)] = \mathbb{E}_{X \sim f_X(x; \theta_0)}[l(\theta, X)] = \int l(\theta; x)f_X(x; \theta_0) \space dx,$$

where $f_X$ is the probability density of $X$, parametrised by $\theta_0$, and where I've omitted the range of integration.

As a point of emphasis, within this frequentist context, you are entirely correct in identifying that we do not assume that $\theta_0$ is in any way a random variable. Nor at any stage would it be coherent to consequently speak of a density function on $\theta_0$ with respect to which we are computing expectations.

In contrast with the widely used notation $\mathbb{E}_X[g(X)] = \mathbb{E}_{X \sim f_X(x)}[g(X)]$, where $f_X$ is again a density, this can take a bit of getting used to, but as with many issues concerning notation, it's really a matter of getting used to what authors are trying to say.

In the context of asymptotic results involving the maximum likelihood estimator, the use of the subscript $\theta_0$ in the expectations operator means the following,

$$\mathbb{E}_{\theta_0}[l(\theta; X)] = \mathbb{E}_{X \sim f_X(x; \theta_0)}[l(\theta, X)] = \int l(\theta; x)f_X(x; \theta_0) \space dx,$$

where $f_X$ is the probability density of $X$, parametrised by $\theta_0$, and where I've omitted the range of integration.

As a point of emphasis, within this frequentist context, you are entirely correct in identifying that we do not assume that $\theta_0$ is in any way a random variable. Nor at any stage would it be coherent to consequently speak of a density function on $\theta_0$ with respect to which we are computing expectations.

In contrast with the widely used notation $\mathbb{E}_X[g(X)] = \mathbb{E}_{X \sim f_X(x)}[g(X)]$, where $f_X$ is again a density, the notation $\mathbb{E}_{\theta_0}[\cdot]$ can take some getting used to, as is the case with most issues concerning notation.

The use of the subscript $\theta_0$ in the expectations operator means the following,

$$\mathbb{E}_{\theta_0}[l(\theta; X)] = \mathbb{E}_{X \sim p(x; \theta_0)}[l(\theta, x)] = \int l(\theta; x)p(x; \theta_0) \space dx,$$

where I've omitted the range of integration.

In the context of asymptotic results involving the maximum likelihood estimator, the use of the subscript $\theta_0$ in the expectations operator means the following,

$$\mathbb{E}_{\theta_0}[l(\theta; X)] = \mathbb{E}_{X \sim f_X(x; \theta_0)}[l(\theta, X)] = \int l(\theta; x)f_X(x; \theta_0) \space dx,$$

where $f_X$ is the probability density of $X$, parametrised by $\theta_0$, and where I've omitted the range of integration.

As a point of emphasis, within this frequentist context, you are entirely correct in identifying that we do not assume that $\theta_0$ is in any way a random variable. Nor at any stage would it be coherent to consequently speak of a density function on $\theta_0$ with respect to which we are computing expectations.

In contrast with the widely used notation $\mathbb{E}_X[g(X)] = \mathbb{E}_{X \sim f_X(x)}[g(X)]$, where $f_X$ is again a density, this can take a bit of getting used to, but as with many issues concerning notation, it's really a matter of getting used to what authors are trying to say.