Derivative Of Bounded Variation Function Is Lebesgue Integrable

The Lebesgue Integrability of the Derivative of a Bounded Variation Function

This article explores the fundamental relationship between functions of bounded variation and Lebesgue integration, specifically proving that the derivative of a function of bounded variation is Lebesgue integrable. Understanding this connection is crucial in real analysis and has significant applications in various fields, including probability theory and Fourier analysis. This proof requires a solid grasp of measure theory and Lebesgue integration.

Understanding the Concepts:

Before diving into the proof, let's define the key terms:

• Function of Bounded Variation (BV): A function f defined on a closed interval [a, b] is of bounded variation if the total variation of f on [a, b], denoted by V_a^b(f), is finite. The total variation is defined as the supremum of the sums Σ_i=1ⁿ |f(x_i) - f(x_i-1)| over all partitions {x₀, x₁, ..., x_n} of [a, b]. Intuitively, this means the function doesn't oscillate too wildly.

• Lebesgue Integrability: A function is Lebesgue integrable if its absolute value is integrable according to the Lebesgue integral. The Lebesgue integral is a generalization of the Riemann integral, capable of integrating a wider class of functions.

• Derivative: We'll be considering the derivative in the sense of Lebesgue's differentiation theorem, which states that for an increasing function, the derivative exists almost everywhere.

The Proof:

The proof relies on several key lemmas and theorems. We won't provide rigorous proofs for all of these, as that would significantly extend this article's length, but we'll outline the main steps and ideas.

1. Decomposition into Increasing Functions: Any function f of bounded variation can be decomposed into the difference of two increasing functions, f = g - h, where g and h are increasing functions. This decomposition is a direct consequence of the definition of total variation.

2. Derivatives of Increasing Functions: Lebesgue's differentiation theorem states that if g is an increasing function, then its derivative g' exists almost everywhere and is Lebesgue integrable. Furthermore, ∫_a^b g'(x) dx ≤ g(b) - g(a).

3. Linearity of the Lebesgue Integral: The Lebesgue integral is a linear operator. This means that if f and g are Lebesgue integrable functions, and c is a constant, then ∫(f + g) = ∫f + ∫g and ∫(cf) = c∫f.

4. Applying the Decomposition: Since f = g - h, and g and h are increasing functions, their derivatives g' and h' exist almost everywhere and are Lebesgue integrable. Therefore, f' = g' - h' exists almost everywhere.

5. Integrability of the Derivative: Because the Lebesgue integral is linear, the derivative of f, f' = g' - h', is also Lebesgue integrable. This is because the difference of two Lebesgue integrable functions is Lebesgue integrable. Moreover, ∫_a^b |f'(x)| dx ≤ V_a^b(f).

Conclusion:

We have outlined the crucial steps in proving that the derivative of a function of bounded variation is Lebesgue integrable. The proof hinges on the decomposition of a BV function into increasing functions, the Lebesgue differentiation theorem for increasing functions, and the linearity of the Lebesgue integral. This result highlights the powerful interplay between the concepts of bounded variation and Lebesgue integration, forming a cornerstone in advanced calculus and analysis. The bound on the integral of the absolute value of the derivative further emphasizes the connection between the total variation and the Lebesgue integral of the derivative.