Sigma-Algebra, Measures, Properties of Measures

Let m be a sigma-algebra, M_1 and M_2 are measures on m. a). Is M = M_1 + M_2 a measure? b). Is M = M_1 - M_2 a measure? c). Is M = M_1M_2 a measure? Either prove or disprove by providing a counter example.

Borel-measurable function

Prove that the following function is Borel-measurable function. f_n(t) = { [t*2^n]*2^-n , 0 < t < n, n , t > or = to n | f_n(t) - t | < 2^-n , t < n } I want a detailed proof. I want to know what one needs to check when proving some function is Borel-measurable function. There are no typos, the way the function is defined for 0 < t < n is approximation for the function

Explanation of the condition - not independent of the Jacobians of functions.

Real Analysis Jacobians (VII) Explanation of the condition - not independent of the Jacobians of functions.

Explanation of the condition - not independent of the Jacobians of functions.

Real Analysis Jacobians (VIII) Explanation of the condition - not independent of the Jacobians of functions.

Description of the working of the Jacobian with trigonometric functions.

Real Analysis Jacobians (IX) Description of the working of the Jacobian with trigonometric functions.