Lebesgue Measure is a mathematical concept used to assign a size or volume to sets in a way that extends the idea of length, area, and volume. It is particularly useful in real analysis and probability theory, allowing for the measurement of more complex sets than traditional methods, such as Riemann integration, can handle. The Lebesgue Measure can measure sets in n-dimensional space, making it possible to define the size of irregular shapes and even some infinite sets. It provides a foundation for the Lebesgue Integral, which is essential for integrating functions that may not be well-behaved under traditional integration methods.