\* (a) Cho E là một dàn phân phối và f f f
f ( x ) + f ( y ) = f ( sup ( x , y ) ) + f ( inf ( x , y ) ) f(x) + f(y) = f(\sup (x, y)) + f(\inf (x, y)) f ( x ) + f ( y ) = f ( sup ( x , y )) + f ( inf ( x , y )) với mọi x x x y y y
f ( sup ( I ) ) + ∑ 2 n ≤ Card ( I ) ( ∑ H ⊂ I , Card ( H ) = 2 n f ( inf ( H ) ) ) = ∑ 2 n + 1 ≤ Card ( I ) ( ∑ H ⊂ I , Card ( H ) = 2 n + 1 f ( inf ( H ) ) )
f(\sup (I))+
\sum_{2n\leq \operatorname{Card}(I)}
\left(
\sum_{\substack{H\subset I,\ \operatorname{Card}(H)=2n}}
f(\inf (H))
\right)
=
\sum_{2n+1\leq \operatorname{Card}(I)}
\left(
\sum_{\substack{H\subset I,\ \operatorname{Card}(H)=2n+1}}
f(\inf (H))
\right)
f ( sup ( I )) + 2 n ≤ Card ( I ) ∑ H ⊂ I , Card ( H ) = 2 n ∑ f ( inf ( H )) = 2 n + 1 ≤ Card ( I ) ∑ H ⊂ I , Card ( H ) = 2 n + 1 ∑ f ( inf ( H )) (Bằng quy nạp theo Card (I).) \*
(b) Đặc biệt, cho A là một tập hợp, cho ( B i ) i ∈ I (B_i)_{i\in I} ( B i ) i ∈ I B i B_i B i B H = ⋂ i ∈ H B i B_H=\bigcap_{i\in H}B_i B H = ⋂ i ∈ H B i
Card ( B ) + ∑ 2 n ≤ Card ( I ) ( ∑ Card ( H ) = 2 n Card ( B H ) ) = ∑ 2 n + 1 ≤ Card ( I ) ( ∑ Card ( H ) = 2 n + 1 Card ( B H ) ) .
\operatorname{Card}(B)+
\sum_{2n\leq \operatorname{Card}(I)}
\left(
\sum_{\operatorname{Card}(H)=2n}
\operatorname{Card}(B_H)
\right)
=
\sum_{2n+1\leq \operatorname{Card}(I)}
\left(
\sum_{\operatorname{Card}(H)=2n+1}
\operatorname{Card}(B_H)
\right).
Card ( B ) + 2 n ≤ Card ( I ) ∑ Card ( H ) = 2 n ∑ Card ( B H ) = 2 n + 1 ≤ Card ( I ) ∑ Card ( H ) = 2 n + 1 ∑ Card ( B H ) . Không có.