Question

Let (X, Σ, μ) be a finite measure space. If f is Σ-measurable, consider the sets Eₙ = f(x). Show that f ∈ L₁ if and only if ∑ from n=1 to [infinity] nμ(Eₙ) is finite. More generally, f ∈ Lₚ for 1 ≤ p < [infinity] if and only if ∑ from n=1 to [infinity] nᵖμ(Eₙ) is finite.

Answer 1

If f is Σ-measurable, the sets Eₙ = f(x). The function f ∈ L₁ if and only if the sum of the measures of the sets Eₙ is finite.

Step-by-step explanation:

In order to show that f ∈ L₁, we need to show that the sum of the measures of the sets Eₙ = f(x) is finite. Let's assume that f ∈ L₁. This means that the absolute value of f is integrable.

By using the definition of measurable functions, we can prove that the sets Eₙ are measurable. Since the measure space is finite, the measure of each set Eₙ is also finite. In other words, μ(Eₙ) < ∞. Therefore, the sum of the measures of the sets Eₙ is finite.

Conversely, if the sum of the measures of the sets Eₙ is finite, then all the sets Eₙ have finite measures. This means that the function f is integrable, and therefore, f ∈ L₁.