/-- A version of Hölder with multiple arguments -/
theorem lintegral_prod_norm_pow_le {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
    (s : Finset ι) {f : ι → α → ℝ≥0∞} (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
    {p : ι → ℝ} (hp : ∑ i in s, p i = 1) (h2p : ∀ i ∈ s, 0 ≤ p i) :
    ∫⁻ a, ∏ i in s, f i a ^ p i ∂μ ≤ ∏ i in s, (∫⁻ a, f i a ∂μ) ^ p i := by
  induction s using Finset.induction generalizing p
  case empty =>
    simp at hp
  

Proof state:
α✝ : Type u_1
inst✝¹ : MeasurableSpace α✝
μ✝ : Measure α✝
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i)) →
    ∀ {p : ι → ℝ},
      ∑ i in s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i in s, f i a ^ p i ∂μ ≤ ∏ i in s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i)
p : ι → ℝ
hp : ∑ i in insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
⊢ ∫⁻ (a : α), ∏ i in insert i₀ s, f i a ^ p i ∂μ ≤ ∏ i in insert i₀ s, (∫⁻ (a : α), f i a ∂μ) ^ p i

Next tactic:
case insert i₀ s hi₀ ih =>
    rcases eq_or_ne (p i₀) 1 with h2i₀|h2i₀
    · simp [hi₀]
      have h2p : ∀ i ∈ s, p i = 0 := by
        simpa [hi₀, h2i₀, sum_eq_zero_iff_of_nonneg (fun i hi ↦ h2p i <| mem_insert_of_mem hi)]
          using hp
      calc ∫⁻ a, f i₀ a ^ p i₀ * ∏ i in s, f i a ^ p i ∂μ
          = ∫⁻ a, f i₀ a ^ p i₀ * ∏ i in s, 1 ∂μ := by
            congr! 3 with x
            apply prod_congr rfl fun i hi ↦ by rw [h2p i hi, ENNReal.rpow_zero]
        _ ≤ (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * ∏ i in s, 1 := by simp [h2i₀]
        _ = (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * ∏ i in s, (∫⁻ a, f i a ∂μ) ^ p i := by
            congr 1
            apply prod_congr rfl fun i hi ↦ by rw [h2p i hi, ENNReal.rpow_zero]
    · have hpi₀ : 0 ≤ 1 - p i₀ := by
        simp_rw [sub_nonneg, ← hp, single_le_sum h2p (mem_insert_self ..)]
      have h2pi₀ : 1 - p i₀ ≠ 0 := by
        rwa [sub_ne_zero, ne_comm]
      let q := fun i ↦ p i / (1 - p i₀)
      have hq : ∑ i in s, q i = 1 := by
        rw [← Finset.sum_div, ← sum_insert_sub hi₀, hp, div_self h2pi₀]
      have h2q : ∀ i ∈ s, 0 ≤ q i :=
        fun i hi ↦ div_nonneg (h2p i <| mem_insert_of_mem hi) hpi₀
      calc ∫⁻ a, ∏ i in insert i₀ s, f i a ^ p i ∂μ
          = ∫⁻ a, f i₀ a ^ p i₀ * ∏ i in s, f i a ^ p i ∂μ := by simp [hi₀]
        _ = ∫⁻ a, f i₀ a ^ p i₀ * (∏ i in s, f i a ^ q i) ^ (1 - p i₀) ∂μ := by
            simp [← ENNReal.prod_rpow_of_nonneg hpi₀, ← ENNReal.rpow_mul,
              div_mul_cancel (h := h2pi₀)]
        _ ≤ (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * (∫⁻ a, ∏ i in s, f i a ^ q i ∂μ) ^ (1 - p i₀) := by
            apply ENNReal.lintegral_mul_norm_pow_le
            · exact hf i₀ <| mem_insert_self ..
            · exact s.aemeasurable_prod <| fun i hi ↦ (hf i <| mem_insert_of_mem hi).pow_const _
            · exact h2p i₀ <| mem_insert_self ..
            · exact hpi₀
            · apply add_sub_cancel'_right
        _ ≤ (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * (∏ i in s, (∫⁻ a, f i a ∂μ) ^ q i) ^ (1 - p i₀) := by
            gcongr -- behavior of gcongr is heartbeat-dependent, which makes code really fragile...
            exact ih (fun i hi ↦ hf i <| mem_insert_of_mem hi) hq h2q
        _ = (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * ∏ i in s, (∫⁻ a, f i a ∂μ) ^ p i := by
            simp [← ENNReal.prod_rpow_of_nonneg hpi₀, ← ENNReal.rpow_mul,
              div_mul_cancel (h := h2pi₀)]
        _ = ∏ i in insert i₀ s, (∫⁻ a, f i a ∂μ) ^ p i := by simp [hi₀]