theorem sum_prob_mem_Ioc_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) {K : ℕ} {N : ℕ}
    (hKN : K ≤ N) :
    ∑ j in range K, ℙ {ω | X ω ∈ Set.Ioc (j : ℝ) N} ≤ ENNReal.ofReal (𝔼[X] + 1) := by
  let ρ : Measure ℝ := Measure.map X ℙ
  haveI : IsProbabilityMeasure ρ := isProbabilityMeasure_map hint.aemeasurable
  have A : ∑ j in range K, ∫ _ in j..N, (1 : ℝ) ∂ρ ≤ 𝔼[X] + 1 :=
    calc
      ∑ j in range K, ∫ _ in j..N, (1 : ℝ) ∂ρ =
          ∑ j in range K, ∑ i in Ico j N, ∫ _ in i..(i + 1 : ℕ), (1 : ℝ) ∂ρ := by
        apply sum_congr rfl fun j hj => ?_
        rw [intervalIntegral.sum_integral_adjacent_intervals_Ico ((mem_range.1 hj).le.trans hKN)]
        intro k _
        exact continuous_const.intervalIntegrable _ _
      _ = ∑ i in range N, ∑ j in range (min (i + 1) K), ∫ _ in i..(i + 1 : ℕ), (1 : ℝ) ∂ρ := by
        simp_rw [sum_sigma']
        refine' sum_bij' (fun (p : Σ _ : ℕ, ℕ) _ => (⟨p.2, p.1⟩ : Σ _ : ℕ, ℕ)) _ (fun a _ => rfl)
          (fun (p : Σ _ : ℕ, ℕ) _ => (⟨p.2, p.1⟩ : Σ _ : ℕ, ℕ)) _ _ _
        · rintro ⟨i, j⟩ hij
          simp only [mem_sigma, mem_range, mem_Ico] at hij
          simp only [hij, Nat.lt_succ_iff.2 hij.2.1, mem_sigma, mem_range, lt_min_iff, and_self_iff]
        · rintro ⟨i, j⟩ hij
          simp only [mem_sigma, mem_range, lt_min_iff] at hij
          simp only [hij, Nat.lt_succ_iff.1 hij.2.1, mem_sigma, mem_range, mem_Ico, and_self_iff]
        · rintro ⟨i, j⟩ hij; rfl
        · rintro ⟨i, j⟩ hij; rfl
      _ ≤ ∑ i in range N, (i + 1) * ∫ _ in i..(i + 1 : ℕ), (1 : ℝ) ∂ρ := by
        apply sum_le_sum fun i _ => ?_
        simp only [Nat.cast_add, Nat.cast_one, sum_const, card_range, nsmul_eq_mul, Nat.cast_min]
        refine' mul_le_mul_of_nonneg_right (min_le_left _ _) _
        apply intervalIntegral.integral_nonneg
        · simp only [le_add_iff_nonneg_right, zero_le_one]
        · simp only [zero_le_one, imp_true_iff]
      _ ≤ ∑ i in range N, ∫ x in i..(i + 1 : ℕ), x + 1 ∂ρ := by
        apply sum_le_sum fun i _ => ?_
        have I : (i : ℝ) ≤ (i + 1 : ℕ) := by
          simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]
        simp_rw [intervalIntegral.integral_of_le I, ← integral_mul_left]
        apply set_integral_mono_on
        · exact continuous_const.integrableOn_Ioc
        · exact (continuous_id.add continuous_const).integrableOn_Ioc
        · exact measurableSet_Ioc
        · intro x hx
          simp only [Nat.cast_add, Nat.cast_one, Set.mem_Ioc] at hx
          simp [hx.1.le]
      _ = ∫ x in (0)..N, x + 1 ∂ρ := by
        rw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]
        · norm_cast
        · exact (continuous_id.add continuous_const).intervalIntegrable _ _
      _ = ∫ x in (0)..N, x ∂ρ + ∫ x in (0)..N, 1 ∂ρ := by
        rw [intervalIntegral.integral_add]
        · exact continuous_id.intervalIntegrable _ _
        · exact continuous_const.intervalIntegrable _ _
      _ = 𝔼[truncation X N] + ∫ x in (0)..N, 1 ∂ρ := by
        rw [integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]
      _ ≤ 𝔼[X] + ∫ x in (0)..N, 1 ∂ρ :=
        (add_le_add_right (integral_truncation_le_integral_of_nonneg hint hnonneg) _)
      _ ≤ 𝔼[X] + 1 := by
        refine' add_le_add le_rfl _
        rw [intervalIntegral.integral_of_le (Nat.cast_nonneg _)]
        

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X
hnonneg : 0 ≤ X
K N : ℕ
hKN : K ≤ N
ρ : Measure ℝ := Measure.map X ℙ
this : IsProbabilityMeasure ρ
⊢ ∫ (x : ℝ) in Set.Ioc 0 ↑N, 1 ∂ρ ≤ 1

Next tactic:
simp only [integral_const, Measure.restrict_apply', measurableSet_Ioc, Set.univ_inter,
          Algebra.id.smul_eq_mul, mul_one]