Lebesgue integrals differ in several key respects from Riemann integrals.  Intuitively, Riemann integrals involve partitioning the domain of a function into disjoint intervals<p>(a_1,a_2),  (a_2,a_3), ... (a_{n-1},a_n)<p>and approximately the area under the graph of the function via sums of the form<p>\sum_i f(x_i) (a_{i+1} - a_i)<p>where x_i is a point in the interval (a_i,a_{i+1}).<p>Lebesgue integrals turn this procedure on its head by partitioning the range of f into disjoint intervals<p>(a_0,a_1), (a_1, a_2), ... (a_{n-1},a_n)<p>and approximating the area under the graph of the function via<p>\sum_i  m({x : a_i &lt; f(x}) &lt; a_{i+1}) a_i<p>where m(E) refers to the &quot;measure&quot; of the set E.  That is,
for each i, we multiply the &quot;size of the  set on which f is mapped to a value near a_i&quot; by a_i and then sum over i.<p>The principle advantage of the Lebesgue scheme is that f can be very badly behaved and the quantities involved are still well-defined and make sense, whereas the Riemann integral only leads to reasonable approximations
if f is somewhat well-behaved (more-or-less continuous).  Otherwise, the value of f(x_i) (x_{i+1}-x_i) is not a reasonable approximation of the area under the graph of f &quot;over the interval (x_{i+1}-x_i)&quot;.<p>There are even more general notions of integral. To my knowledge, most are based on observing that an integral is a linear functional on some space which should satisfy certain properties.