Fundamental Theorem of Calculus à la Lebesgue

• categories: Measure Theory, Real Analysis, Theorem

Fundamental Theorem of Calculus à la Lebesgue

• Statement: Let $f\in L^1([a,b])$ be a Lebesgue integrable function, and define its cumulative distribution function $F(x)$ by:

  $$F(x)={\int }_a^{x}f(t)dt$$

  for $x\in [a,b]$. Then:

  1. Absolute Continuity: $F(x)$ is absolutely continuous on $[a,b]$.
  2. Differentiability: $F^{\prime }(x)=f(x)$ almost everywhere on $[a,b]$.

• Intuition: This result generalizes the classical Fundamental Theorem of Calculus to Lebesgue integrable functions. The derivative of the cumulative integral $F(x)$ exists almost everywhere and equals the original function $f$, even if $f$ is not continuous or pointwise defined everywhere.

• Key Properties:

  1. $F(x)$ is well-defined and absolutely continuous due to $f\in L^1([a,b])$.
  2. $F^{\prime }(x)=f(x)$ holds almost everywhere, meaning that the points where the derivative fails to exist form a set of measure zero.

• Proof Outline:

  1. Cumulative Function: The definition of $F(x)$ as an integral of $f$ ensures that $F$ is an absolutely continuous function.
  2. Lebesgue Differentiation Theorem: Apply the Lebesgue Differentiation Theorem to $F^{\prime }(x)$ to show that the derivative of $F$ equals $f$ almost everywhere.