I was reading a paper that define the kernel integral operator as follows:
We define the kernel integral operator $\mathcal{K}$. Let $\kappa^{(l)} \in C(D \times D; \mathbb{R}^{d_{l+1} \times d_l})$ and let $\nu$ be a Borel measure on $D$. Then we define $\mathcal{K}$ by $$(\mathcal{K}v_l)(x) = \int_D \kappa^{(l)}(x, y)v_l(y)d\nu(y) \qquad \forall x \in D.$$
What does the kernel integral operator do? What is $C(D \times D; \mathbb{R}^{d_{l+1} \times d_l})$? What is a Borel measure?
To answer what a Borel measure is in a concise way, one has to make use of a lot of concepts in measure theory.
Let $D$ be a topological space, with topology $\tau$. This is the collection of all open sets of $D$. The $\sigma$-algebra on $D$ generated by $\tau$ is called the Borel $\sigma$-algebra. A Borel measure is a measure on this $\sigma$-algebra.
The notation $C(D\times D; \mathbb{R}^{d_{\ell+1}\times _{\ell}})$ usually means the space of continuous functions $f: D \times D \to \mathbb{R}^{d_{\ell+1}\times d_{\ell}}$. Here, the topology on $D \times D$ is the product topology.
The idea of an integral kernel is that it defines an integral operator. Roughly speaking, one can define an operator over some function space by integrating each element of this function space across some well-behaved function. In other words: $$T: \varphi \mapsto (T\varphi)(x) := \int k(x,y)\varphi(y)dy$$ provided these objects are well-defined and understood.
Intuitively, it is the "functional" or "continuous" version of a matrix multiplication operator. Given a vector $\varphi \in \mathbb{R}^{N}$, an $N\times N$ matrix $M = (M_{ij})$ can be viewed as an operator: $$M: \varphi \mapsto (M\varphi)_{i} := \sum_{ij}M_{ij}\varphi_{j}$$ In this simple case, the indices are just positive integers. An integral operator plays the role of the matrix when the "indices become continuous variables". This is, of course, just pictoric, but is the intuition used by physicist most of the time when they use integral operators.