Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence...

TL;DR

Lesbesgue integrable functions are defined as measurable functions for which the integral of the absolute value is finite. The Lesbesgue integral of a function over a measurable set is defined as the integral of its positive part minus the integral of its negative part.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: All right, so last time we defined the integral of a non-negative measurable function, the Lesbesguen rule. Now we are going to define the Lesbesguen rule for a general class, a more general class of functions. So Lesbesgue integrable functions. So what does this mean? Let E be a measurable subset of R, ... Read More

Key Insights

• ⚾ Lesbesgue integrable functions are defined based on the finiteness of the integral of their absolute value.
• 🥳 The Lesbesgue integral of a function over a measurable set is defined in terms of the integrals of its positive and negative parts.
• ❓ Integrable functions satisfy properties such as linearity, additivity, and positivity.
• 👻 The dominated convergence theorem allows for the computation of integrals in the limit of a sequence of functions.

Questions & Answers

Q: How is a Lesbesgue integrable function defined?

A function is Lesbesgue integrable if the integral of its absolute value is finite.

Q: What is the Lesbesgue integral of a function over a measurable set?

The Lesbesgue integral of a function over a measurable set is defined as the integral of its positive part minus the integral of its negative part.

Q: What are some properties of integrable functions?

Integrable functions satisfy properties such as linearity, additivity, and positivity. They can be split into positive and negative parts, and their integrals can be computed separately.

Q: What is the dominated convergence theorem?

The dominated convergence theorem states that if a sequence of measurable functions is dominated by an integrable function and converges pointwise to a limit function, then the limit function is integrable and the limit of the integrals equals the integral of the limit function.

Q: Is the Lesbesgue integral of a continuous function equal to its Riemann integral?

Yes, the Lesbesgue integral of a continuous function on a closed and bounded interval is equal to its Riemann integral.

Summary & Key Takeaways

• Lesbesgue integrable functions are measurable functions for which the integral of the absolute value is finite.

• The Lesbesgue integral of a function over a measurable set is defined as the integral of its positive part minus the integral of its negative part.

• Properties of integrable functions include linearity, additivity, and positivity.

• The dominated convergence theorem states that if a sequence of measurable functions is dominated by an integrable function and converges pointwise to a limit function, then the limit function is integrable and the limit of the integrals equals the integral of the limit function.

• The Lesbesgue integral of a continuous function on a closed and bounded interval is equal to its Riemann integral.