YES, the absolute value of an integral is always less than the integral of the absolute value for Lebesgue integrals.
An integral is like a sum, but fancier. So look at a sum 1+(-5)+2+(-3)=-5. Take the absolute value, |1+(-5)+2+(-3)|=5. But, if you take the absolute value of the individual terms and then add them, you get |1|+|-5|+|2|+|-3|=1+5+2+3=11. 11>5
The reason is that when you add them all up and then take the absolute value, the positive and negative terms of the sum get to cancel each other out and make the absolute value of the sum greater overall. When you take the absolute value of individual terms, you force them all to "point in the same direction" (positive) so they are working together and the sum gets higher.
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