Lebesgue theorem

The average amount by which a function in $L^1(\mathbb{R})$ differs from its values is small almost everywhere on small intervals.¹

Note

Let $f\in L^1(\mathbb{R})$. Then

$$\underset{t\downarrow 0}{\lim }\frac{1}{2t}{\int }_{b-t}^{b+t}|f(x)-f(b)|dx=0$$

for almost evert $b\in \mathbb{R}$.

Note L^1 function with a continuous, we can since they are dense\dots \square

(Sketch) The theorem is obvious for continuous functions (bound with the supremum). The strategy is to approximate any

Footnotes §

1. This statement and the main guide of this note is the book MIRA. ↩