The 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 measure theory, allowing for the measurement of more complex sets than traditional methods, such as those defined by intervals or simple geometric shapes. Developed by the mathematician Henri Léon Lebesgue in the early 20th century, the Lebesgue measure can handle sets that are not easily measurable, such as those with irregular boundaries. It provides a foundation for integration, enabling the calculation of areas under curves and the analysis of functions in a rigorous manner.