Only from the meaning of the formula, Sxf(x) dx converges (so mathematics expects to calculate the value of conditional convergence. )

But "expectation" imposes the condition of absolute convergence of the sequence Sxf(x) dx, because mathematical expectation is often calculated by sampling from the population.

According to the theorem of large numbers and the central limit theorem, when the number of samples extracted from the population is large, the arithmetic mean of sample values is interesting and total.

Expectation (of course, I mean discrete and continuous, which can be understood similarly), because sampling is random, it is calculated from sampling in the population.

The general expectation requires that the sum of series Sxf(x) dx cannot be changed due to the change of the order of terms, and this requirement should also be met for integrals.

Sxf(x) dx should not change its sum because of the change of the order of the terms (for example, the staggered series converges, but its even or odd terms may not necessarily converge).

It is also required to be absolutely convergent.

Therefore, mathematical expectation requires the absolute convergence of Sxf(x) dx, which will inevitably lead to the convergence of Sxf(x) dx and the existence of mathematical expectation. Ji Gu

The convergence of number Sxf(x) dx is a necessary and sufficient condition for the expected existence.