Lebesgue integration is a mathematical concept that extends the idea of integration beyond traditional methods, such as Riemann integration. It allows for the integration of a wider class of functions by focusing on the measure of the set where the function takes certain values, rather than just the intervals on the x-axis. In Lebesgue integration, functions are measured in terms of their "size" or "volume" over a given set. This approach is particularly useful in probability theory and real analysis, as it can handle functions that may be discontinuous or not well-behaved, providing a more flexible framework for integration.