The Lebesgue Integral is a mathematical concept used to extend the idea of integration beyond traditional methods, such as the Riemann Integral. It allows for the integration of a wider class of functions by focusing on the values that the function takes and the measure of the sets where these values occur, rather than just the intervals on the x-axis. In the Lebesgue Integral, functions are integrated with respect to a measure, which quantifies the size of sets in a more flexible way. This approach is particularly useful in probability theory and real analysis, as it can handle functions that may be discontinuous or defined on complex domains.