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

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