Lecture 9: Lebesgue Measurable Functions

TL;DR

Measurable functions are closed under pointwise limits and can be defined on sets of measure 0.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: All right. So last lecture, we concluded our discussion about measurable functions, I mean, measurable sets. And remember our original motivation was that we're trying to build an integral that somehow surpasses the properties of that of the Riemann integral in that, hopefully, this larger class of... Read More

Key Insights

• ❓ Measurable functions are motivated by the desire to create an integral that surpasses the limitations of the Riemann integral.
• 😚 Measurable functions are defined as functions where the inverse image of closed intervals is measurable.
• 😚 Measurable functions are closed under taking pointwise limits.

Questions & Answers

Q: What is the motivation for defining measurable functions?

Measurable functions are motivated by the desire to create an integral that has properties beyond the Riemann integral, such as being closed under pointwise limits.

Q: What is the definition of a measurable function?

A measurable function is a function where the inverse image of closed intervals is measurable.

Q: Can measurable functions be closed under taking pointwise limits?

Yes, measurable functions are closed under taking pointwise limits, meaning that the limit of a sequence of measurable functions is also measurable.

Q: Can a measurable function be modified on a set of measure 0 and still remain measurable?

Yes, a measurable function can be modified on a set of measure 0 and still remain measurable. The modification will not affect the measurability of the function.

Summary & Key Takeaways

• Measurable functions are motivated by the desire to create an integral that surpasses the limitations of the Riemann integral.

• Measurable functions are defined as functions where the inverse image of closed intervals is measurable.

• Measurable functions can be closed under taking pointwise limits and can be modified on sets of measure 0 while still remaining measurable.