I was reading the paper "The Logistic Normal Distribution for Bayesian, Nonparametric, Predictive Densities" published in 1988 in Journal of the American Statistical Association by Peter J. Lenk.
At the beginning of the paper, the author mentions a sigma-algebra that I don't understand. His goal is to work with random pdf afterwards and he states that for a probability space $(\Omega, F, P)$ and a space $X$:
[A random pdf] $f$ is modelled by a stochastic process from $\Omega \times X$ to $\mathbb R^+$, the positive reals such that the sample paths integrate to $1$. With this goal in mind, $\mathbb {R^{+}}^X$ is the space of functions from $X$ to $\mathbb R^+$ with the Borel sigma-field $B(X, \mathbb R^+)$.
I don't understand what the Borel sigma-field mentionned is supposed to be, as the topology considered was not specified. Is there a "standard"/"canonical" choice of topology that allows to define this sigma-algebra, or is it an imprecision in the paper ?
Thank you very much in advance!
As you say, no special topological hypothesis is made concerning $X$, so one must assume that by $B(S,\Bbb R^+)$ the author intends the product $\sigma$-field on $(\Bbb R^+)^X$. As the author intends to integrate $f(\omega, y)$ with respect to $y$ and thereby obtain $\mathcal F$-measurable functions like $\omega\mapsto\int_A f(\omega,y)d\lambda(y)$, one really needs (to use Fubini) to assume that $(\omega,y)\mapsto f(\omega,y)$ is is $\mathcal F\otimes\mathcal X$-measurable, where $\mathcal X$ is the $\sigma$-field on $X$ upon which $\lambda$ is defined. This implies that for each Borel subset $B\subset\Bbb R^+$ and each $y\in X$, $$ \{\omega\in\Omega: f(\omega,y)\in B\} $$ is $\mathcal F$-measurable. This is enough to guarantee that $\omega\mapsto f(\omega,\cdot)$, viewed as a map from $\Omega$ to $(\Bbb R^+)^X$ is $\mathcal F/B(S,\Bbb R^+)$-measurable.