Improper Integral - Improper Riemann Integrals and Lebesgue Integrals

Improper Riemann Integrals and Lebesgue Integrals

In some cases, the integral

can be defined as an integral (a Lebesgue integral, for instance) without reference to the limit

but cannot otherwise be conveniently computed. This often happens when the function f being integrated from a to c has a vertical asymptote at c, or if c = ∞ (see Figures 1 and 2). In such cases, the improper Riemann integral allows one to calculate the Lebesgue integral of the function. Specifically, the following theorem holds (Apostol 1974, Theorem 10.33):

• If a function f is Riemann integrable on for every b ≥ a, and the partial integrals

are bounded as b → ∞, then the improper Riemann integrals

both exist. Furthermore, f is Lebesgue integrable on [a, ∞), and its Lebesgue integral is equal to its improper Riemann integral.

For example, the integral

can be interpreted alternatively as the improper integral

or it may be interpreted instead as a Lebesgue integral over the set (0, ∞). Since both of these kinds of integral agree, one is free to choose the first method to calculate the value of the integral, even if one ultimately wishes to regard it as a Lebesgue integral. Thus improper integrals are clearly useful tools for obtaining the actual values of integrals.

In other cases, however, a Lebesgue integral between finite endpoints may not even be defined, because the integrals of the positive and negative parts of f are both infinite, but the improper Riemann integral may still exist. Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits. For example,

cannot be interpreted as a Lebesgue integral, since

But is nevertheless Riemann integrable between any two finite endpoints, and its integral between 0 and ∞ is usually understood as the limit of the Riemann integral: