General Functions

Finally, we extend the definition to any measurable function $f$.

General measurable functions

Definition: Integrable Function

Let $f$ be any measurable function. We call $f$ integrable if

$$\int |f| \, d\mu < \infty$$

Decomposition

We can decompose any function $f$ into its positive and negative parts:

$$f^+(x) = \max\{f(x), 0\} \quad \text{(Positive part)}$$ $$f^-(x) = -\min\{f(x), 0\} = \max\{-f(x), 0\} \quad \text{(Negative part)}$$

Note that both $f^+$ and $f^-$ are non-negative measurable functions (so Step 3 applies to them). Key identities:

• $f = f^+ - f^-$
• $|f| = f^+ + f^-$

Definition of the integral

Definition

For an integrable function $f$, we define the Lebesgue integral as:

$$\int f \, d\mu = \int f^+ \, d\mu - \int f^- \, d\mu$$

Condition for integrability

Since $|f| = f^+ + f^-$, by linearity (Property 3 for non-negative functions):

$$\int |f| \, d\mu = \int f^+ \, d\mu + \int f^- \, d\mu$$

Thus, $f$ is integrable if and only if both $\int f^+ \, d\mu < \infty$ and $\int f^- \, d\mu < \infty$.

Properties

Exercise 8

Problem

Prove that Properties 1-6 also hold for general integrable functions.

Hint

Use the decomposition $f = f^+ - f^-$ and the linearity of the integral for non-negative functions. Be careful with signs!