19th century is an era of highly developed creativity and rigor in the history of mathematics. The establishment of complex variable function theory and the rigor of analysis, the advent of non-Euclidean geometry and the perfection of projective geometry, and the birth of group theory and noncommutative algebra are typical mathematical achievements in this century. The new ideas they contained had a far-reaching impact on mathematics in the 20th century.

At the beginning of the 20th century, Cantor's set theory was formally accepted as a branch of mathematics, and on this basis, measure theory and integral theory were developed. In particular, Leberg's integral theory had a decisive influence on the later development of real function theory, and was applied to harmonic analysis, differential equations and later functional analysis. Lebesgue integral has been popularized in various ways in more than ten years, especially Ladon integral, which unifies Stirges integral and Lebesgue integral, and has important application to later integral geometry and even X-ray imaging theory. When Joinet invented the general process, he proved the reciprocity of differential and integral for irreconcilable derivatives, thus obtaining the most general concept of integral. At the end of 19, Bell's research on function classes was combined with point set theory, which led to the emergence of analytic set theory. Set theory was greatly developed in the Soviet Union and Poland.