If μ is a probability measure on α, s is a convex closed set in E, and f is an integrable function sending μ-a.e. points to s, then the expected value of f belongs to s: ∫ x, f x ∂μ ∈ s. See also convex.sum_mem for a finite sum version of this lemma.

If μ is a non-zero finite measure on α, s is a convex closed set in E, and f is an integrable function sending μ-a.e. points to s, then the average value of f belongs to s: ⨍ x, f x ∂μ ∈ s. See also convex.center_mass_mem for a finite sum version of this lemma.

If μ is a non-zero finite measure on α, s is a convex closed set in E, and f is an integrable function sending μ-a.e. points to s, then the average value of f belongs to s: ⨍ x, f x ∂μ ∈ s. See also convex.center_mass_mem for a finite sum version of this lemma.

If μ is a non-zero finite measure on α, s is a convex set in E, and f is an integrable function sending μ-a.e. points to s, then the average value of f belongs to closure s: ⨍ x, f x ∂μ ∈ s. See also convex.center_mass_mem for a finite sum version of this lemma.

Jensen's inequality: if a function g : E → ℝ is convex and continuous on a convex closed set s, μ is a finite non-zero measure on α, and f : α → E is a function sending μ-a.e. points to s, then the value of g at the average value of f is less than or equal to the average value of g ∘ f provided that both f and g ∘ f are integrable. See also convex_on.map_center_mass_le for a finite sum version of this lemma.

Jensen's inequality: if a function g : E → ℝ is concave and continuous on a convex closed set s, μ is a finite non-zero measure on α, and f : α → E is a function sending μ-a.e. points to s, then the average value of g ∘ f is less than or equal to the value of g at the average value of f provided that both f and g ∘ f are integrable. See also concave_on.le_map_center_mass for a finite sum version of this lemma.

Jensen's inequality: if a function g : E → ℝ is convex and continuous on a convex closed set s, μ is a finite non-zero measure on α, and f : α → E is a function sending μ-a.e. points of a set t to s, then the value of g at the average value of f over t is less than or equal to the average value of g ∘ f over t provided that both f and g ∘ f are integrable.

Jensen's inequality: if a function g : E → ℝ is concave and continuous on a convex closed set s, μ is a finite non-zero measure on α, and f : α → E is a function sending μ-a.e. points of a set t to s, then the average value of g ∘ f over t is less than or equal to the value of g at the average value of f over t provided that both f and g ∘ f are integrable.

Jensen's inequality: if a function g : E → ℝ is convex and continuous on a convex closed set s, μ is a finite non-zero measure on α, and f : α → E is a function sending μ-a.e. points of a set t to s, then the value of g at the average value of f over t is less than or equal to the average value of g ∘ f over t provided that both f and g ∘ f are integrable.

Jensen's inequality: if a function g : E → ℝ is concave and continuous on a convex closed set s, μ is a finite non-zero measure on α, and f : α → E is a function sending μ-a.e. points of a set t to s, then the average value of g ∘ f over t is less than or equal to the value of g at the average value of f over t provided that both f and g ∘ f are integrable.

Jensen's inequality: if a function g : E → ℝ is convex and continuous on a convex closed set s, μ is a probability measure on α, and f : α → E is a function sending μ-a.e. points to s, then the value of g at the expected value of f is less than or equal to the expected value of g ∘ f provided that both f and g ∘ f are integrable. See also convex_on.map_center_mass_le for a finite sum version of this lemma.

Jensen's inequality: if a function g : E → ℝ is concave and continuous on a convex closed set s, μ is a probability measure on α, and f : α → E is a function sending μ-a.e. points to s, then the expected value of g ∘ f is less than or equal to the value of g at the expected value of f provided that both f and g ∘ f are integrable.

If f : α → E is an integrable function, then either it is a.e. equal to the constant ⨍ x, f x ∂μ or there exists a measurable set such that μ t ≠ 0, μ tᶜ ≠ 0, and the average values of f over t and tᶜ are different.

If an integrable function f : α → E takes values in a convex set s and for some set t of positive measure, the average value of f over t belongs to the interior of s, then the average of f over the whole space belongs to the interior of s.

If an integrable function f : α → E takes values in a strictly convex closed set s, then either it is a.e. equal to its average value, or its average value belongs to the interior of s.

Jensen's inequality, strict version: if an integrable function f : α → E takes values in a convex closed set s, and g : E → ℝ is continuous and strictly convex on s, then either f is a.e. equal to its average value, or g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ.

Jensen's inequality, strict version: if an integrable function f : α → E takes values in a convex closed set s, and g : E → ℝ is continuous and strictly concave on s, then either f is a.e. equal to its average value, or ⨍ x, g (f x) ∂μ < g (⨍ x, f x ∂μ).

If E is a strictly convex normed space and f : α → E is a function such that ‖f x‖ ≤ C a.e., then either this function is a.e. equal to its average value, or the norm of its average value is strictly less than C.

If E is a strictly convex normed space and f : α → E is a function such that ‖f x‖ ≤ C a.e., then either this function is a.e. equal to its average value, or the norm of its integral is strictly less than (μ univ).to_real * C.

If E is a strictly convex normed space and f : α → E is a function such that ‖f x‖ ≤ C a.e. on a set t of finite measure, then either this function is a.e. equal to its average value on t, or the norm of its integral over t is strictly less than (μ t).to_real * C.