ENNReal.eventually_le_limsup

theorem ENNReal.eventually_le_limsup : ∀ {α : Type u_1} {f : Filter α} [inst : CountableInterFilter f] (u : α → ENNReal), ∀ᶠ (y : α) in f, u y ≤ Filter.limsup u f

The theorem `ENNReal.eventually_le_limsup` states that for any type `α`, any filter `f` on `α` that is a countable intersection filter, and any function `u` from `α` to the extended nonnegative real numbers (denoted as `ENNReal`, which includes all nonnegative real numbers plus positive infinity), there exists an event in `f` such that for all elements `y` of `α` in this event, the value `u(y)` is less than or equal to the limit superior (`limsup`) of `u` with respect to the filter `f`. In mathematical terms, we can express this as: ∀y ∈ α, eventually for `f`, it holds that `u(y) ≤ limsup f(u)`. This is a property that connects the notions of limit superior and filters in the context of extended nonnegative real numbers.

More concisely: For any countable intersection filter `f` on type `α` and function `u` from `α` to `ENNReal`, there exists an element `y` in `α` such that for all `x` in `α` in the event `f` containing `y`, `u(x)` is less than or equal to the limit superior of `u` with respect to `f`.