$\lim\limits_{\delta_j\downarrow0}\int\limits_{\delta_j\le x\le 1}K(x)$ exists $\iff \lim\limits_{\delta_j\downarrow0}\int\limits_{\delta_j\le x}K(x)\phi(x)=\langle W,\phi\rangle$ with $\phi$ is a test function and $W$ is a tempered distribution.
First; How is it possible that the integral exists if $K$ is defined away from $0$ ?
If LHS exists (call it $L$) then RHS is equal to $\int\limits_{|x|\ge 1}K(x)\phi(x)+\int\limits_{|x|\le 1 }K(x)[\phi(x)-\phi(0)]dx+\phi(0)L$. If the left integral can be bounded by $C\sup\limits_{x\in\mathbf R^n}(1+|x|)^N\lvert\phi(x)\rvert$, is this sufficient for the implication left to right ?
What is the behaviour of a test function (or is it called a Schwarz function ?) near $0$ ? Googling didn't help, It shall have compact support and decay faster than an inverse of any polynomial