Calculating the integral in measure theory

Question

I'm very new to measure theory, so I don't really understand how to do this: $\int_Axe^yd\lambda^2$ for A:=$\left\{(x,y)\in R : 0\le x\le 1, 0\le y\le \frac1 2 x^2\right\}$

Do I calculate it using regular integral rules? I'm sorry if this sounds like a I don't know anything question, but I am truly lost and I hope that someone can still help out, thanks in advance!

Answer 1

Assuming $\lambda^2$ is supposed to be the Lebesgue measure on $\mathbb R^2$, you can simply treat this integral as a Riemann integral, since $(x,y)\mapsto x\cdot e^y$ is continuous and therefore Riemann integrable. In this case, the Riemann integral and Lebesgue integral are equal, see here.

Now, the Riemann integral can be written as a double integral of the form $$\int_ *^*\int_*^*xe^y dxdy$$ You'll just have to figure out the integration boundaries $(*)$.

Answer 2

Here is a theorem that you might find handy:

Let $f \colon \varOmega \to \mathbb R$ be measurable and $f \geq 0$ almost everywhere. Then \begin{equation} \int f \, d\lambda = \int_0^{\infty} \lambda\left(\{f \geq t\}\right) \, dt \end{equation}