黎曼积分,有不可积函数,而勒贝格积分规定,处处稠密的有理数域的测度为0,而原本同样是处处稠密的实数域或无理数域的测度才不为0.这种“规定”唯一“正当”的理由,就是通常被认为是“不可数”的实数域或无理数域是连续的(连续统),因此才有非零的测度值.而“可数”的有理数域被认为是不连续的,所以测度只能为0.本文经分析指出,此种理由不能成立.而在笔者提出的“增量分析”的观点下,测度不过就是增量本身,因此使理论更加合理自然,也更加简化.黎曼积分,使传统微积分中不可积的一些函数可积了,而增量分析,使黎曼积分中一些测度为0的积分域可以不为0了.对微积分求导及连续统的可数性问题,均在前期大量工作的基础上,又提出更加有力的全新的观点,使相关理论更为简洁、明确,达到了无可置疑的程度.%Riemann integral,product function,and lebesgue integral regulation,everywhere populated measure of rational number field to 0,and originally is the same everywhere populated measure of real number field or irrational number field is 0.The only justification for this "stipulation" is that the real or irrational fields,which are generally considered to be "uncountable",are continuous (continuous),hence the non-zero measure values.The rational domain of "countable" is considered to be discontinuous,so the measure can only be 0.The analysis indicates that this reason cannot be established.In the view of "incremental analysis" proposed by the author,the measure is only the increment itself,so the theory is more reasonable and natural,and more simplified.The Riemann integral,which can integrate some functions that are not integrable in traditional calculus,and the incremental analysis,is that some of the integrals in the Riemann integral are 0.For the derivation calculus and continuum,countability in early,on the basis of a lot of work,and put forward a more powerful new point of view,and make the related theory more concise,clear,to the degree to which there is no doubt.
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