I'm wondering if there exists some variant of Lebesgue's dominated convergence theorem, which applies to a sequence of functions that is defined over a changing sequence of spaces. In particular, I'm interested in the following examle.
Let $f_1,f_2,\ldots$, be a sequence of functions, such that $f_d:\mathbb{R}^d\rightarrow\mathbb{R}$, $d\in\mathbb{N}^*$. Denote by $\cdot^{(d)}$ the restriction of a sequence to its first $d$ elements. Suppose for every sequence of reals $x=x_1,x_2,\ldots$, it holds that $$ \lim_{d\rightarrow\infty}f_d\left(x^{(d)}\right)=0, $$ and that there exists a sequence of functions $g_1,g_2,\ldots$, with $g_d:\mathbb{R}^d\rightarrow\mathbb{R}$, and $M>0$, such that for all $d\in\mathbb{N}^*$ $$ \left|f_d\left(x^{d}\right)\right|\leq g_d\mbox{ and } \int_{x^{(d)}\in\mathbb{R}^d}g_d\left(x^{(d)}\right)\mathrm{d}x^{(d)}\leq M. $$ Can we show that $$ \lim_{d\rightarrow\infty}\int_{x^{(d)}\in\mathbb{R}^d}f_d\left(x^{(d)}\right)\mathrm{d}x^{(d)}=0? $$ If false, can we add extra conditions for this to hold (e.g., the functions are continuous and/or or restricted to $[a,b]^d$, ...)?
I would really appreciate some hints or references.