A function f: X → ℝ is measurable if preimages of Borel sets are measurable (in the σ-algebra of X). This is the natural notion of 'integrable function' in measure theory. Measurable functions are closed under pointwise limits and composition with continuous functions.
In calculus you integrated continuous functions — they're "nice" enough that slicing the domain into intervals and summing works without issue. Measure theory asks a more fundamental question: what is the largest class of functions you can meaningfully integrate? The answer is measurable functions. The definition might seem circular at first — a function is measurable if it "plays well" with the σ-algebra — but there's a concrete intuition: a function is measurable if you can always ask "where does f take values in this set?" and get a measurable set as the answer.
Precisely, f: (X, 𝒜) → ℝ is measurable if for every Borel set B ⊆ ℝ, the preimage f⁻¹(B) = {x ∈ X : f(x) ∈ B} belongs to the σ-algebra 𝒜 on X. You've already studied the Borel σ-algebra on ℝ — it's generated by open intervals. A useful simplification: you only need to check preimages of sets of the form (-∞, a) or equivalently {x : f(x) < a}, since these generate the Borel sets. If this family of preimages all lands in 𝒜, the full definition is satisfied.
The measurability condition connects directly to your measure space prerequisites. On the domain side, you have a σ-algebra 𝒜 telling you which subsets of X are "observable." On the range side, the Borel σ-algebra tells you which subsets of ℝ are "observable." Measurability requires that f translate observable events on the range back to observable events on the domain. This is exactly the structure needed to define the Lebesgue integral: to integrate f, you ask "how much measure does the set {x : f(x) ∈ [a, a+da]} have?" — and you need that set to be measurable to assign it a measure.
The closure properties make measurable functions a workable class. Any continuous function is measurable (since preimages of open sets are open, hence Borel, hence measurable). Sums, products, and scalar multiples of measurable functions are measurable. Most importantly, pointwise limits of measurable functions are measurable: if fₙ → f pointwise and each fₙ is measurable, then so is f. This property is what gives the Lebesgue integral its power over the Riemann integral — you can take limits of functions and stay within the integrable class, enabling theorems like the Dominated Convergence Theorem that allow you to pass limits through integrals freely.