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Its been almost 3 weeks that I've asked this question on math.SE and I haven't recieved a single comment/answer. Also I could not find any information on this question in some text books I refered. Hence I request you to kindly peruse the question and suggest if its suitable to migrate it to mathoverflow in the hope of getting some useful comments/answers.
why did you write \delta for the upper limit of the integral in the definition of DI? Don't you mean \infty?
(Also, allow me to encourage you to use your real name while posting on meta and mathoverflow.)
He is probably wanting the limit to be independent of of the upper limit $\delta$, which it will be so long as $g$ is locally integrable.