Nonstandard Analysis And It's Partial Applications In Fractal Geometry

The paper consists of five parts. The first part describes the prior knowledge of nonstandard analysis, including the logical statement, three principles of nonstandard analysis, and the compactness theorem. The second part expands the classical mathematics of the real number field to the hyper real number field, that is to say, increases infinity and infinitesimal on the basis of he primary number, and introduces the nature of the hyper real number field as well as it's application in the classical mathematics. The third part deduces the non-standard equivalent definition of the funetion limit at some point by equivalent definitions of series convergence and limit, and introduces it's continuity and application in nonstandard analysis. The fourth part shows nonstandard equivalent definition of the calculus. In the last part,utilizing the method of nonstandard analysis, we prove that:(1)for any near standard interior subset A and a standard Borel regular finite outer measure μ of standard complete metric space (X, d), if f is a nonnegative real function in S0(X), then the integral inequality holds, in special case°(μ(A))<μ(°A) for f=1;(2) if f∈S0(X) and E is a s-compact subset of X, then where v(*) denote the shadow of*. Moreover, by using these results we give a judgment method of fractal with the interior nonempty and a calculating method for integration with respect to a fractal function on the Hausdorff measure space, the efficiency of the two methods is shown with examples.