For applications in physics and other sciences the Lebesgue integral is not critical. Even if you care about making sure your calculations are mathematically sound then the theory of the Riemann integral (much older than Lebesgue&#x27;s) is sufficient.<p>However in mathematics, it did change how people think about integration. &quot;Integral&quot; now usually means &quot;Lebesgue integral&quot; outside of specialized applications. One of its advantages is that a lot of theorems involving integration become simpler to state, since with the Riemann integral you need to add more conditions to make sure it is defined.