let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for A being set st A in S holds
0 <= M . A
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for A being set st A in S holds
0 <= M . A
let M be sigma_Measure of S; :: thesis: for A being set st A in S holds
0 <= M . A
let A be set ; :: thesis: ( A in S implies 0 <= M . A )
reconsider E = {} as Element of S by PROB_1:4;
assume A in S ; :: thesis: 0 <= M . A
then reconsider A = A as Element of S ;
M . E <= M . A by MEASURE1:31, XBOOLE_1:2;
hence 0 <= M . A by VALUED_0:def 19; :: thesis: verum