Since the invertor of the saxophone was from Belgium, how did the saxophone become so popular in jazz music? I mean, it was created by a European in the 1840's, it wasn't created by Americans at the end of the 19th century...
Why does any instrument exist in any style of music? Because musicians at the time have access to those instruments and want to use them.
Jazz began in the city of New Orleans, Louisiana, in the United States of America, roughly around the year 1900. I cannot find any reference to saxophones being involved in jazz in New Orleans during the first decade of jazz. However, the clarinet was involved in jazz from the start. As you know, the saxophone family of instruments were designed to be a more modern and easier-to-play substitute for the older clarinet family of instruments. So wherever a clarinet-family instrument is used, saxophones may be substituted.
The earliest jazz made use of various instruments including the tuba, clarinet, trumpet, trombone, cornet, piano, upright bass, drum kit, and the banjo. Of those instruments only the banjo and the drum kit (practically speaking) were invented in the USA. All of the rest of those instruments as we now know them were invented or developed in Europe in earlier centuries, just as the saxophone was. So you might ask why jazz includes any of those other instruments as well. The answer is "because they do."
The first uses of the saxophones in jazz seems to have been around the year 1912 and might have taken place outside of New Orleans, such as in Ohio or Kansas City. As this was already 75 years after the invention of the saxophones, they were already commonplace throughout the USA. Saxophones were already in use and quite popular in symphony orchestras and wind bands, notably in the work of the great American composer John Philip Sousa and his military band music, going back to 1893.
Saxophones were an instrument in the polka bands all across the USA. Polka first appeared in the USA around the 1840s, the same time as the invention of the saxophone in Belgium. Polka was brought to the USA by immigrants from places including Poland, Czechoslovakia, and Germany. Polka bands could be found in places as diverse as northern Mexico, southern Texas and up to Chicago, Illinois and Milwaukee, Wisconsin and many places inbetween.
In summary, before jazz was invented, the United States of America had symphony orchestras, wind orchestras, military bands, and polka bands, and saxophones were commonplace. When jazz appeared, it used the instruments that were already there, and the saxophones were among them.
My brief and over-simplified definition of jazz is this: Jazz is what resulted when people in the USA who were descended from slaves imported from Africa became free in New Orleans, and obtained access to not only Western European musical instruments but also exposure to the Western European musical tradition and the White musicians who played it. Jazz is unique to the USA; it is neither African nor European, but something that evolved out of the musical traditions and the materials of the two. It could only have happened in the USA at a particular place and time.
Saxophones can be effectively expressive of emotion in music, express anger, joy, confusion, melancholy, a wide spectrum of sound effects and more. It was a natural fit for just about any kind of jazz.