This is the description on math.stackexchange.com for Functional Analysis:
Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modelled by these vector spaces.
Whoever wrote that tag description did a fairly decent job of describing Functional Analysis. There are some basic ingredients: Infinite-dimensional spaces, topology that allows approximation, continuous operators and continuous linear functionals. Lebesgue integration is nice in such settings, but not necessary. However, general operator theory on function spaces is not so easily studied without Measure Theory.
An introductory course may focus on metric topology, inner product spaces, and normed linear spaces. The classical integration spaces using Riemann integration do not give you complete spaces, but you can define an abstract completion where the original spaces are densely embedded. The completion will end up being a Lebesgue space, but you don't necessarily need to know that in order to work with the abstract completion. You can study continuous functionals, dual spaces, continuous linear operators, spectral theory on complete spaces, as well as operator algebras, without digging into the specifics of Lebesgue spaces, but the examples won't be as thoroughly investigated without Measure Theory. However, the general theory will be just as powerful and meaningful without such details, and classical ordinary differential operators can be effectively studied using only Riemann integration, including the existence of orthogonal bases of functions and integral transforms such as the Fourier transform.
A more advanced course will typically deal with topological vector spaces, which are vector spaces equipped with general topologies that give continuous vector and scalar multiplication operations. The study of more general spaces requires General Topology and convex analysis. Examples include spaces of distributions and other advanced topics in Partial Differential Equations. Sobolev spaces and embeddings can be effectively studied if you know Lebesgue integration. Advanced versions of functional calculus can be studied in this setting.