The most intuitive way of teaching the two integration to kids is to use the following diagram:
For Riemann integration, the diagram directly translates to: $\lim_{n\to\infty}\sum_{i=1}^nf(x_i)(x_{i+1}-x_{i})$ and we use this to teach the K-12 kids all the time. It is quite useful.
Let $y=f(x)$
Why don't we use the similar formula to teach the Lebesgue integration:
$$\lim_{n\to\infty}\sum_{i=1}^n \mu(y>y_{i})(y_{i+1}-y_i)?$$
I've never seen this infinite sum anywhere. All texts use some $f_n$ and take the limit.