# Is every coordinate-wise marginal of a proper joint distribution also proper?

Suppose that we have a probability density function $$\pi(x_1, \ldots, x_n)$$ which is the density of a vector-valued random variable $$X$$ in $$\mathbb{R}^n$$. Here the density is proper, i.e., $$\int_{\mathbb{R}^n} \pi(x_1, \ldots, x_n) dx_1 \ldots dx_n = 1 < \infty$$.

Without any additional restrictions on the joint density, does this imply that any coordinate-wise marginal density of the form $$\pi(x_i) = \int \pi(x_1, \ldots, x_n) dx_1 \ldots dx_{i-1} dx_{i+1} \ldots dx_n$$ is also proper, i.e., that $$\int_{\mathbb{R}} \pi(x_i) dx_i = 1 < \infty$$ for any $$1 \leq i \leq n$$?

The motivation for this question is that I am considering a hierarchical Bayesian model involving an improper prior, and although I know that the posterior density over all model parameters is proper, I have seemingly found a marginal that is not proper (although this may be an error in my analytic computation). Intuitively, I would expect that any marginal of a proper joint should also be proper?

The same notations should not be used for different entities, so let me define $$\pi_i(x_i) = \int_{\mathbb R^{n-1}} \pi(x_1, \ldots, x_n) dx_1 \ldots dx_{i-1} dx_{i+1} \ldots dx_n$$ as the $$i$$-th marginal. Then $$\int_{\mathbb{R}} \pi_i(x_i) dx_i = \int_{\mathbb{R}}\int_{\mathbb R^{n-1}} \pi(x_1, \ldots, x_n) dx_1 \ldots dx_{i-1} dx_{i+1} \ldots dx_n dx_i\\ =\int_{\mathbb R^{n}} \pi(x_1, \ldots, x_n) dx_1 \ldots dx_{i-1} dx_i dx_{i+1} \ldots dx_n = 1$$ by Fubini's Theorem.