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The cumulative distribution function F(x) = P(X ≤ x) uniquely determines the distribution of X. A probability density function f is the Radon-Nikodym derivative when the distribution is absolutely continuous with respect to Lebesgue measure. Distributions may also be purely singular or have atoms.
From your prerequisite on random variables as measurable functions, you know that a random variable X is a measurable map X: (Ω, ℱ) → (ℝ, ℬ(ℝ)), and you've worked with the Borel σ-algebra ℬ(ℝ). The distribution of X is the pushforward measure μ_X defined by μ_X(B) = P(X ∈ B) = P(X⁻¹(B)) for Borel sets B ⊆ ℝ. Everything you want to know about X — probabilities, expectations, quantiles — is encoded in μ_X. Crucially, μ_X is a probability measure on (ℝ, ℬ(ℝ)), so it inherits all the properties of measures: countable additivity, σ-finiteness on ℝ, and so on.
The cumulative distribution function F(x) = P(X ≤ x) = μ_X((−∞, x]) is a convenient scalar summary of the distribution. Its properties follow directly from measure theory: F is non-decreasing (measures of nested sets are nested), right-continuous (by the continuity-from-above property of measures), and satisfies F(−∞) = 0, F(+∞) = 1. Conversely, any function with these three properties is the CDF of some random variable. The key structural fact is that F uniquely determines μ_X — there is a one-to-one correspondence between CDFs satisfying these properties and probability measures on ℝ. This is why CDFs are the universal language for describing distributions.
A probability density function arises when the distribution is absolutely continuous with respect to Lebesgue measure λ. Absolute continuity μ_X ≪ λ means: whenever a Borel set has Lebesgue measure zero, it also has μ_X-measure zero. When this holds, the Radon-Nikodym theorem guarantees a unique (up to a.e. equivalence) measurable function f such that μ_X(B) = ∫_B f dλ for all Borel B. This function f is the PDF, and it equals F'(x) almost everywhere. The Radon-Nikodym framing is the rigorous version of the intuitive statement "probability = area under the density curve" — you are literally saying that the probability measure is absolutely continuous with respect to length (Lebesgue measure), and the derivative f is the density of probability per unit length.
Not all distributions have PDFs. A discrete distribution concentrates all its mass on a countable set of points: P(X = xₖ) = pₖ > 0 for a countable collection. Such a distribution is singular with respect to Lebesgue measure — it is entirely supported on a set of Lebesgue measure zero — so no Radon-Nikodym density exists. A stranger beast is the purely singular continuous distribution, whose CDF is continuous (no atoms) but is constant almost everywhere. The Cantor distribution is the canonical example: its CDF is the Devil's staircase, which increases from 0 to 1 entirely on the Cantor set (measure zero), so its "density" would have to be zero almost everywhere but still integrate to 1 — a contradiction, so no density exists. The full Lebesgue decomposition theorem says every distribution uniquely decomposes into an absolutely continuous part (has a PDF), a discrete part (atoms), and a singular continuous part.
Understanding the Radon-Nikodym perspective matters because it unifies otherwise disparate formulas. The PDF formula P(a < X ≤ b) = ∫_ab f(x) dx and the PMF formula P(X = k) = pₖ are not really two different things — they are both special cases of P(X ∈ B) = ∫_B dμ_X, one where μ_X is absolutely continuous (use Radon-Nikodym to get f) and one where μ_X is a sum of point masses. Characteristic functions, the joint distribution theory that builds on this topic, and all of measure-theoretic probability depend on fluency with this framework.