# Show Lebesgue integration by Lebesgue integral definition.

The question is as following, by using Lebesgue integral definition, show that $$\int_{[0,1]}fdm=\frac{1}{2}$$ where $$f(x) = x$$ on $$[0,1]$$. Here is my approach.

For any $$\epsilon>0$$ and $$A=[0,1]$$, choose $$\frac{1}{2}<\delta<\frac{\epsilon}{m(A)}$$ such that whenever $$||P||<\delta$$, we have

$$|L(f,P)-\frac{1}{2}|=|\sum_{i=1}^{n}y_i^*.m(\{x\in A|y_{i-1}\leq f(x) as required.

Am I correct with this proof ?

• You have proved Riemann integrability. I haven't seen any body defining Lebesgue integral in this fashion. You have used the tags measure theory and lebegue-integral but have you studied Lebesgue intargation? – Kavi Rama Murthy Jun 25 at 5:53
• Isn't this Lebesgue integration ?? I used measurable set to prove it . – Ling Min Hao Jun 25 at 12:48