This image contains a mathematical example that relates to the properties of measurable functions and integrals over a measurable set.

Breakdown of the content:

Statement:

The example says:

• If $f$ is a measurable function,
• $m(E)$ (the measure of the set $E$) is finite, $m(E) < \infty$,
• There are two constants $A \leq f(x) \leq B$ on the set $E$, for almost every $x \in E$,

Then, the inequality holds: $m(E) \cdot A \leq \int_E f \, dx \leq m(E) \cdot B$

This inequality is commonly known as an application of the sandwich theorem or squeeze theorem in integration. It bounds the integral of the function $f$ between the lower and upper bounds of $f$, multiplied by the measure of the set $E$.

Solution:

• $A \chi_E$ and $B \chi_E$ are integrable functions, where $\chi_E$ is the characteristic (or indicator) function of the set $E$, which equals 1 on $E$ and 0 elsewhere.
• Since both these functions are integrable, Theorem 9(iv) is applied. This theorem likely establishes conditions under which integrable functions maintain their integrability properties under certain operations.

The final sentence mentions that this leads to the main result of the section, implying that this is a stepping stone for proving a larger theorem related to integration and measurable functions.

Key Takeaways:

• The function $f$ is measurable and bounded by constants $A$ and $B$ on a set $E$.
• The integral of $f$ over $E$ is bounded by the values of $f$ at its extremes, scaled by the measure of $E$.
• Indicator functions $\chi_E$ help establish the integrability of these bounds.

Would you like a more detailed breakdown of a specific part or further explanation of any concept?

Here are 5 related questions to expand on this:

1. What are the conditions under which a function is considered measurable?
2. How does the sandwich theorem apply to integrals in other contexts?
3. What is the characteristic function $\chi_E$, and why is it useful in integration?
4. How does Theorem 9(iv) relate to this result (you may need to refer to it)?
5. What does it mean for a function to be integrable over a set?

Tip: Understanding measurable sets and functions is essential for studying more advanced topics in real analysis and Lebesgue integration.