Proving Lebesgue Integration Yields Same Answer As Riemann

Question

I am working on extended essay for my International Baccalaureate Programme Full Diploma, and I chose mathematics as my topic. My research questions is: “What are the shortcomings of Riemann integration, why do they exist, and how do other forms of integration overcome those limitations?”

The first alternative I will discuss is Lebesgue integration. By starting with a graphical analysis of a bijective function $f\left( x\right)$ whose limit will be evaluated from $x=a$ to $x=b$, I have attempted to write the process as a unified equation,

$$L=\small\lim_{n\to\infty}\sum_{i=1}^{n}\left(\left[ f^{-1}\left( a+i\left[\dfrac{f\left( b\right) -f\left( a\right)}{n}\right]\right)-f^{-1}\left( a+\left[ i-1\right]\left[\dfrac{f\left( b\right) -f\left( a\right)}{n}\right]\right)\right]\left( a+i\left[\dfrac{f\left( b\right) -f\left( a\right)}{n}\right]\right)\right)$$

By comparison, my formula for Riemann integration follows the form

$$R=\lim_{n\to\infty}\sum_{i=1}^{n}f\left( a+\frac{i\left( b-a\right)}{n}\right)\left(\frac{ b-a}{n}\right)$$

I am having extreme difficulty proving that $L=R$, since I do not know how to algebraically evaluate the limit of a series. However, I did try to test my formulae for $f\left( x\right)=x^2+1$, $a=0$ and $b=a$. While the formula $R$ has successfully yielded the correct answer under every trial, I have only gotten $L$ to converge to $R$ when integrating symmetrical trigonometric functions; in this case, however, $L$ ended up appearing to be undefined for all $x>0$.

Does anyone have any suggestions? Have I made a key mathematical error? an arithmetical one? I pulled my information from Arturo Magidin's answer under “Lebesgue Integration Basics.”

Thank you for any help you can give.

Answer 1

I don't know where you're getting that formula for Lebesgue integration. Here's how I would show that the Lebesgue integral agrees with the Riemann integral for Riemann integrable functions.

1. First look at step functions. Show it's true there.
2. If $f$ is Riemann integrable on $[a,b]$, then there are sequences of step functions $g_n$ and $h_n$ such that $g_n \le f \le h_n$ and $\int_a^b h_n(x)\; dx$ and $\int_a^b g_n(x)\; dx$ both converge to the Riemann integral $\int_a^b f(x)\; dx$ as $n \to \infty$.
3. Use this to show that $f$ is Lebesgue integrable and that this same limit is its Lebesgue integral.