Measurable Functions — Preimage Explorer
Function f: [0,1] → ℝ
Continuous: f(x) = sin(2πx)
Step: f(x) = 0 if x<0.5, 1 if x≥0.5
Dirichlet: f(x) = 1 if x∈ℚ, 0 otherwise
Non-measurable: 1_V (Vitali set)
Target Borel set B ⊂ ℝ
(0.2, 0.8) — open interval
[0.3, 0.7] — closed interval
(0, ∞) — half-line
ℚ ∩ [0,1] — countable set
ℝ \ ℚ — uncountable set
Sigma-algebra on domain
Borel σ-algebra (full)
Trivial {∅, X}
x where f(x) ∈ B (preimage)
x where f(x) ∉ B
Non-measurable region
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