Notation for Lebesgue integration theory

Question

I'm currently studying Lebesgue's integration theory and the symbol used in this field to denote an integral is the same as the one used in Riemann's integration theory, the famous $\int$ symbol, and since I used for 7 years the $\int$ symbol to write Riemann integral, I always get confused.

Is there a good notation (good meaning here that I won't get confused if I encouter a new integration theory in the future) to explicitly specify that I am working with a Lebesgue's integral ?

Thank you in advance for (maybe) taking the time to answer this trivial question.

Answer 1

Yes, one very common (used for example in A garden of Integrals by Burk) way to denote different integrals is so prefix them with the letter of the name. So you might have:

• Riemann Integral $R\int$
• Cauchy Integral $C\int$ (not to be confused with the Cauchy principal value often denoted $CP\int$
• Riemann-Stieltjes Integral $\text{R-S}\int$
• Lebesgue Integral $L\int$

and so on for all the other kinds.

Note that for the lebesgue integral it's also quite common to write $d\mu$ or $d\mu(x)$ rather than $dx$ which can be used to differentiate between Riemann and Lebesgue in a nice way.

Answer 2

There are multiple ways of writing the Lebesgue integral. Some times people write $$ \int f \, d\mu $$ $$ \int f \, d \lambda $$ Which means that it is with respect to the Lebesgue measure.