Lebesgue measure and Application?
The previous answer was good, but it is important to point out that the main reason that Lebesgue measure is used is that it has very good properties under various limit processes. For example, if you look at the collection of square integrable functions for Riemann integrals, you do not get a complete Hilbert space (you only get an inner product space), but if you do the same for Lebesgue integrals, you do get a Hilbert space (every Cauchy sequence will converge). This allows the proof of many existence theorems, from solutions to differential equations, to decomposition theorems. This is why quantum mechanics is appropriately done with Lebesgue integrals rather than Riemann.
What should everyone know about Lebesgue Integration?
I think one of the most important things to know about the Lebesgue integral is that it was created to solve some of the flaws of the Riemann integral. Lebesgue wrote a letter to his friend M. Piccard in 1901 explaining some of the problems of the Riemann integral and how his integration method was going to sort this issues.
How does the Lebesgue integration work?
When you perform Riemann integration of a function f(x) say over a interval [0,1], you are dividing the spatial interval [0,1] into pieces (e.g. the x-axis), choosing a point in each piece to "sample" the function, multiplying the sampled values by the lengths of those intervals, and summing all of that up. When you perform Lebesgue integration, you are dividing the "distribution" of values of f into pieces (e.g. the y-axis instead of the x-axis), looking at the frequency with which the function falls into that piece of the distribution (this is usually called the "measure" of that set), multiplying the two together, and then summing it all up. Lebesgue integration isn't necessarily better for solving most problems, since all the integral techniques you learn in Calculus are derived from the Riemann integral. Where Lebesgue integration does become useful is when you are solving theoretical problems because it focuses on the distribution of the values of functions, which makes it conducive toward deriving bounds and inequalities. These last two things are at the heart of a large portion of mathematics.
What is the best book for learning about Lebesgue integration?
I really liked these lecture notes written by William Chen, an emeritus professor at Macquarie University: Introduction to Lebesgue Integration. I used these notes in my first real analysis class, towards the end. I’d start with chapter 4, entitled “The Lebesgue Integral”. Chapters 5 and 6, entitled “Monotone Convergence Theorem” and “Dominated Convergence Theorem”, respectively, are quite good too.One thing that I think these notes do really well is that they start from a bigger picture kind of perspective, introducing Lebesgue integration as a means for solving interesting problems. It’s this kind of “search for motivation” kind of writing that I find really helpful when learning a new subject.Photo Credit: Introduction to Lebesgue Integration
Which relationship occurs between Lebesgue measure and Lebesgue integration?
A measure allows you to assign mass to some sets. Once you can do that, you can integrate functions. ~Hence, to properly define Lebesgue integration, you need to first properly define the Lebesgue measure.~Edit: Blunder on my part; you can actually do Lebesgue integration with any measure, one of which is the Lebesgue measure.
What are the differences between Riemann and Lebesgue integrals?
The Lebesgue integral uses Lebesgue measure which uses open coverings to measure an interval. The sum of lengths of minimal such coverings then provide the measure. This means the L-measure can address functions which are pointwise defined (such as heads or tails in the flip of a coin), as well as functions which are continuous or somewhat continuous. It is, in fact difficult to find a set which is not L-measurable. The existence of such a set is equivalent to the axiom of choice. The Lebesgue integral, and corresponding differentation would be the same as the corresponding Riemann integral for the common functions. If something is Riemann integrable it is L-integrable; however, the other direction does not hold. Most important conclusions coming from the Riemann integral can also be proven for the L-integral, but the L-integral can be applied to more functions. Thus it provides a more powerful tool in addressing the theory of some disciplines.