In the case of n is even, this paper proves the existence of the rare sequence {nk} suchthat the summability takes place at each Lebesgue point satisfying some antipole conditions (see [1-8]).

In this paper, we give a brief poot of absotutely Henstock integ rable is Lebesgue integrable,next we use Lebesgue point to structure the gauge δ and proved that absolutely Henstock integrable is Mcshane integrable.

A subset D of R is called to have property L, if there is a Lebesgue number for every family of open interval covering D.

And quiet a lot of these points exist from the perspective of the Lebesgue measure.

Proved that this kind of open domain does not carry a nontrivial doubling measure, Also constructed a bounded closed Jordan domainΩon R~2,on which the limit of Lebesgue measure is not a doubling measure.

This article is dealing with the problem of the subject context analyzing for Real Variable Function Theory,including Lebesgue measure,integral theory, the developing clue of the problem, the main idea, the main methods of skill, the whole construction, etc .

The estimation of the Lagrange interpolation basic polynomial on the second Chebyshev nodes and the range of the Lebesgue constant are given.

Lebesgue integral and its usage in probability;

The relation of Lebesgue integral and generalized integral;

The equivalent definitions of Lebesgue integral;