for your post! To explain the example from the book, let's first define the statement in the red frame: "If a function f is integrable on a Banach space X, then there exists a sequence of simple functions converging to f in the norm of X."
This statement is essentially saying that any integrable function on a Banach space can be approximated by a sequence of simple functions in the same space. To prove this, we can use a technique called "cutting and pasting," where we divide the integral of f into smaller intervals and approximate each interval with a simple function. Then, by taking the limit of these simple functions, we can show that they converge to f in the norm of X.
In the book, the authors provide an example of how this technique can be applied to a specific function and Banach space. By following their steps and using the definition of integrability and the properties of Banach spaces, we can see how the sequence of simple functions converges to the original function in the given Banach space.
I hope this helps to explain the example in the book. Let me know if you have any further questions or need clarification.
