A basic doubt on Lebesgue integration Can anyone tell me at a high level (I am not aware of measure theory much) about Lebesgue integration and why measure is needed in case of Lebesgue integration? How the measure is used to calculate the horizontal strip mapped for a particular range? 
 A: Imagine a cashier who is in-charge of counting coins at a bank and thereby report the total money collected everyday to the bank authorities. Also, let us assume that the coins can only be of denomination $1$, $2$, $5$ and $10$. Now say he receives the coins in the following order:
$$5,2,1,2,2,1,5,10,1,10,10,5,2,1,2,5,10,2,1,1,1$$
Now he has two different ways to count.
$1$. The first way is to count the coins as and when they come, i.e., he does
$$5+2+1+2+2+1+5+10+1+10+10+5+2+1+2+5+10+2+1+1+1$$
which gives $79$.
$2$. The second way is as follows. He has $4$ boxes, one box for each denomination, i.e., the first box is for coins with denomination $1$, the second box is for coins with denomination $2$, the third box is for coins with denomination $5$ and the last box is for coins with denomination $10$. He drops the coins in the corresponding box as and when it comes. At the end of the day, he counts the coins in each box, i.e., he counts that there are $7$ coins with denomination $1$, $6$ coins with denomination $2$, $4$ coins with denomination $5$ and $4$ coins with denomination $10$. He hence finally reports the total money as
$$7 \times 1 + 6 \times 2 + 4 \times 5 + 4 \times 10 = 79$$
$\color{red}{\text{The first method is the Riemann way of summing}}$ the total money, while $\color{blue}{\text{the second method is the Lebesgue way of summing}}$ the same money.
In the second way, note that there are $4$ sets, i.e., the boxes for denominations $1$, $2$, $5$ and $10$. The measure of each of these sets/boxes is nothing but the denomination of each of these boxes, i.e., the measure of each of these sets is $1$, $2$, $5$ and $10$ respectively and the functional value on each of these sets is nothing but the number of coins in that particular denomination.
A: The idea of (exterior) measure is required to define the integral of the characteristic function of a measurable set. Without this basic building block you cannot even define the Lebesgue integral.
A: This is more about Lebesgue integration in general, and not the the horizontal strip business.
I imagine it something like this: the Riemann integral is only able to approximate functions by rectangles. Rectangles basically only use a single set that we "know" the length of: the interval. It's easy to compute the length of an interval, and so if $\chi_{[a,b]}$ is the function that is 1 on the interval $[a,b]$ and 0 off of it, we can easily compute the integral $$\int c \chi_{[a,b]} = c(b-a).$$ To compute the integrals of other functions, we approximate them by functions like these. This gives the Riemann integral.
But the Riemann integral has a few issues. It doesn't behave well with limits and there are lots of functions that "ought" to be integrable but aren't. So what we do is replace boring intervals $[a,b]$ as our "basic integration set" with a much larger class: measurable sets. Think of these sets as being a very, very large collection of sets that we can find the length of. A measure is an assignment of a number to each of these sets in a way that is compatible with our notion of area. Let's call $\mu(A)$ the measure of a set, given by the measure $\mu$. This is basically the area of $A$, or perhaps the length of $A$.
Now, we think about our function $\chi_A$ again, that takes the value of $1$ on $A$ and $0$ off $A$. If our integral does anything like what is should, then we had better have $$\int \chi_A d\mu = \mu(A).$$ How do we compute the integrals over other functions? We essentially approximate them by finite sums of functions like the ones above and that tells us what the area should be.
There are, of course, lots of details not mentioned here, but in short: the Lebesgue integral allows more flexibility in approximation by letting us approximate by a much richer collection of sets.
