The integral of a function is intuitively the “area under the graph”, but it is now generalized in many ways. And when functions “converge” the function in some sense, do their integrals also converge it? The question “is this true?” can be a yes or no. The answer depends on the situation. By clarifying the nature of the concept of integrals, I am trying to clarify the mechanism by which such things do or do not happen.
Also, probability and integration are closely related. The so-called “expected value” or “variance (standard deviation)” is the integral of a random variable, and stochastic integrals are used to analyze stochastically varying processes such as stock prices.
However, the concept of “probability” that currently prevails in mathematics does not match well enough the image that people have of “probability. For this reason, there is still a debate among philosophers as to what probability is. This also has a negative impact on probability in school education. By resolving this issue, the human way of thinking about probability should evolve.