Let $\left< I, \nu \right>$ be a measure space and that $\{ \left< X_t, \mu_t \right> \}_{t \in I}$ be a family of measure spaces. Let $X = \prod_{t \in I} X_t$ modulo differences on $\nu$-null sets (i.e. $X$ is a set of equivalence classes where two elements belong to the same equivalence class if they only differ for $t$ in a $\nu$-null set). We will typically denote an element of $X$ with $x$ and the value of $x$ at $t \in I$ with $x_t$.
Let $\int^\times$ denote the product integral, i.e. for a $\nu$-measurable function $f : I \to [0, \infty]$ we set $$\int^\times_I f(t)^{d\nu(t)} = \exp \int_I \ln f(t) \, d\nu(t).$$ Here we of course take $\ln 0 = -\infty$, $\ln \infty = +\infty$, $\exp (-\infty) = 0$ and $\exp (+\infty) = \infty$.
Let $\{ f_t : X_t \to [0, \infty] \}_{t \in I}$ be a family of functions and define $f : X \to [0, \infty]$ by $f(x) = \int^\times_I f_t(x_t)^{d\nu(t)}$.
Now define $$\int f(x) \, \mathcal{D}x = \int^\times_I \left( \int f_t(x_t) \, d\mu_t(x_t) \right)^{d\nu(t)} \tag{*}\label{funcint}$$
Note
If $I = \{ 1, \ldots, n \}$ and $\nu$ is the counting measure, then $X$ is the ordinary Cartesian product $X_1 \times \cdots \times X_n,$ $f$ is the tensor product $$f(x_1, \ldots, x_n) = (f_1 \otimes \cdots \otimes f_n)(x_1, \ldots, x_n) = f_1(x_1) \cdots f_n(x_n),$$ and $\int f(x) \, \mathcal{D}x$ is the multidimensional integral $$\int \cdots \int f_1(x_1) \cdots f_n(x_n) \, dx_1 \cdots dx_n = \left( \int f_1(x_1) \, dx_1 \right) \cdots \left( \int f_n(x_n) \, dx_n \right).$$
Questions
- Is the "integral" \eqref{funcint} linear?
- Can it be extended to a linear map on the linear span of all functions of the form $f(x) = \int^\times_I f_t(x_t)^{d\nu(t)}$?