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 measuring the size of the set where the function takes certain values, rather than just summing up areas under curves. Developed by the mathematician Henri Léon Lebesgue in the early 20th century, the Lebesgue integral is particularly useful in real analysis and probability theory. It provides a more flexible framework for dealing with functions that may be discontinuous or defined on complex sets, making it essential for advanced mathematical applications.