Lebesgue integrability is a mathematical concept that is used to determine if a function can be integrated over a given interval. It is based on the Lebesgue measure, which is a way of measuring the size of a set in a more general way than the traditional notion of length, area, or volume.
A step function is a function that takes on a finite number of values over a given interval. They are important in proving Lebesgue integrability because they can be used to approximate more complicated functions and their integrals can be easily calculated.
A sequence of step functions converges uniformly on [0,1] if, for any given error tolerance, there exists a point in the sequence after which all the functions in the sequence are within that error tolerance of each other at every point in the interval [0,1].
The uniform limit of step functions is a key tool in proving Lebesgue integrability because it allows us to approximate more complicated functions with simpler, easily integrable step functions. By taking the limit of the step functions, we can show that the integral of the original function exists and is equal to the integral of the limit of the step functions.
No, it is not necessary for a function to be continuous to be Lebesgue integrable. While continuous functions are Lebesgue integrable, there are also non-continuous functions that are also Lebesgue integrable. The key requirement for Lebesgue integrability is that the function is well-behaved and has a finite integral over the given interval.