I was reading Philip Protter's Stochastic Integration and Differential Equations textbook. He mentions that an operator, $I_X$, induced by $X$ should be linear to be called an integral.
I have a basic understanding of integration concepts and I understand that Lebesgue integral can be proven to be linear. However, I was wondering what is the significance of linearity for integrals. Why did the author say that the operator should be linear to be called an integral. Is it simply because integration is an extension of sums, which is linear? Or is there a more rigorous mathematical reason behind it?
Thanks in advance for any help.