Lebesgue differentiation theorem

The Lebesgue Differentiation Theorem states that for a suitable function, the value of the function at almost every point can be recovered by looking at the average of the function over smaller and smaller regions around that point. As these regions shrink, the average approaches the actual function value, allowing us to understand the pointwise behavior of the function through local averages. This theorem bridges the concepts of pointwise values and average behavior, ensuring that, despite potential irregularities, the local average provides an accurate estimate of the function at nearly all points.