Lebesgue integral,
n. the integral of a measurable function, f, over a subset, E, of a measure space with respect to its measure, μ, written
for example, with respect to Lebesgue measure, the integral of any measurable function, f, over the rationals,
A Lebesgue integral can be constructed by the device of taking the limit of integrals of simple functions approximating the function, and is equal to the Riemann integral
if E is a bounded interval on which f is bounded with discontinuities that form a set of measure 0.
∫E f d μ;
for example, with respect to Lebesgue measure, the integral of any measurable function, f, over the rationals,
∫Q f d μ = 0.
A Lebesgue integral can be constructed by the device of taking the limit of integrals of simple functions approximating the function, and is equal to the Riemann integral
∫E f(x) dx
if E is a bounded interval on which f is bounded with discontinuities that form a set of measure 0.