Does an example of a function exist where the Lebesgue Integral doesn't work?

Question

I am studying the Lebesgue Integral as part of a project and I am convinced the Lebesgue Integral is a more powerful method of integration than that of Bernhard Riemann's method. An example of where this is established to be true is the Dirichlet Function.

However I am wondering if there is an example of a function where the Lebesgue Integral also struggles to define an accurate area under a function?

QUESTION EDITED - 00:22 - 04/04/19

Answer 1

There are lots and lots of non Lebesgue integrable functions. Here are two examples:

Let $A$ be a non measurable set then the characteristic function of $A$ is not Lebesgue integrable.

For a more wild example define $f:[0,1]\to [0,1]$ as follows: If after some term every second number in the decimal expansion of $x$ is recurring (for exampe $0.375389192919593...$) then $f(x)=0.$(the numbers between the recurring digits.) in our case $f(0.375389192919593...)=0.12153....$ If there is no recurrence set $f(x)=0$