How does Lebesgue integral handle functions with non-zero infimum? How do Lebesgue integral handle functions with non-zero intercept? For example, take dyadic coefficients. The $blue$ boxes are summed by$$\sum_ka_k1_{A_k}$$ but what about the area coloured $green$? Only $\emptyset$ is mapped to that part of $A_k$'s so the simple function doesn't give any weight on those $A_k$'s. However to get an "area under curve", one should include that part to preserve the notion of integration.

 A: That's not really how it's done. The idea is to take a small interval $[y,y+\varepsilon]$ and then consider the set $f^{-1}([y,y+\varepsilon])$. This set, which I will denote by $X_y$, consists of those elements that are mapped to the interval $[y,y+\varepsilon]$. So to get an approximation of the area under this part of the curve, one can consider the box with height $y$ and as base the set $X_y$. The area of this box is the simply $y\times \mu(X_y)$ where $\mu(X_y)$ is the 'length' of $X_y$.
To get an approximation of the integral of this function, you can partition the range of the function $f$ into small pieces, and for each peace you can consider the boxes described above. Summing the areas of these boxes approximates the integral. Taking a limit of this procedure yields the Lebesgue integral.
The hard part is defining the proper notion of length. Consider for example the function $$f:\mathbb{R}\rightarrow \mathbb{R}:x\mapsto \begin{cases}1 & \mbox{ if } x\in \mathbb{Q},\\
0 &\mbox{ if } x\notin \mathbb{Q}.
\end{cases}$$
Then $f^{-1}([\frac{1}{2},\frac{3}{2}])=\mathbb{Q}$. So to find the area of the corresponding box, you need to know the length of $\mathbb{Q}$. It's not obvious what this length is.
A: You seem to be misunderstanding how the Lebesgue integral treats simple functions. Think of the simplest simple function: $f:=aI_A$, where $A$ is the unit interval. This function is literally "nothing but green". The Lebesgue integral of this $f$ is by definition $a\cdot \text{Leb}(A)$, which is height times width, which is the area under the curve.
By linearity, you can see that this 'area under the curve' interpretation extends to linear combinations of indicator functions. Then extend to non-negative functions via approximation by simple functions, and onward to arbitrary measurable functions.
