The notations in the question are ambiguous (Bjørn Kjos-Hanssen showed that the other interpretation cannot be correct). I assume that the expression of interest is given by $$ g(t) = \int_0^t \Sigma(s)\,dW(s)\;, $$ where $W$ is a $C(\mathbb{R})$-valued Wiener process with covariance $c$ (at time $1$) and $\Sigma(s) \in C(\mathbb{R})^*$ is the finite measure given by $\Sigma(s) f = \int_{T_1}^{T_2}\sigma(s,u)f(u)\,du$. The natural thing here is to interpret $c$ as the bilinear map on $C(\mathbb{R})^*$ such that, for measure $\mu$ and $\nu$, $$ c(\mu,\nu) = \int c(u,v)\, \mu(du)\,\nu(dv)\;. $$ Itô isometry then indeed reads $$ \mathbb{E} g(t)^2 = \int_0^t \mathbb{E} c(\Sigma(s),\Sigma(s))\,ds \;, $$ assuming of course that $\Sigma$ is adapted and square integrable. Regarding references, any book on SPDEs would do, for example "Stochastic Equations in Infinite Dimensions" by Da Prato & Zabczyk or Section 3 of my lecture notes.

The notations in the question are ambiguous (Bjørn Kjos-Hanssen showed that the other interpretation cannot be correct). I assume that the expression of interest is given by $$ g(t) = \int_0^t \Sigma(s)\,dW(s)\;, $$ where $W$ is a $C(\mathbb{R})$-valued Wiener process with covariance $\hat c$ (at time $1$) and $\Sigma(s) \in C(\mathbb{R})^*$ is the finite measure given by $\Sigma(s) f = \int_{T_1}^{T_2}\sigma(s,u)f(u)\,du$. Here, the covariance $\hat c$ is the bilinear map on $C(\mathbb{R})^*$ such that, for measures $\mu$ and $\nu$, $$ \hat c(\mu,\nu) = \int c(u,v)\, \mu(du)\,\nu(dv)\;. $$ Itô isometry then indeed reads $$ \mathbb{E} g(t)^2 = \int_0^t \mathbb{E} \hat c(\Sigma(s),\Sigma(s))\,ds \;, $$ assuming of course that $\Sigma$ is adapted and square integrable. Regarding references, any book on SPDEs would do, for example "Stochastic Equations in Infinite Dimensions" by Da Prato & Zabczyk or Section 3 of my lecture notes.