Question

To prove that if ∫ fdμ=∫ fdν for all continuous functions f:Rᵏ→R, then μ=ν, you can use the following steps

Answer 1

To prove that if ∫ fdμ=∫ fdν for all continuous functions f:Rᵏ→R, then μ=ν, follow these steps: assume the equality holds for all continuous functions, consider specific functions f(x) = 1 and f(x) = 0, subtract the integrals, simplify, and conclude that μ=ν.

To prove that if ∫ fdμ=∫ fdν for all continuous functions f:Rᵏ→R, then μ=ν, you can use the following steps:

1. Assume that ∫ fdμ=∫ fdν for all continuous functions f:Rᵏ→R.
2. Let's consider a continuous function f:Rᵏ→R with f(x)=1, where x is any point in Rᵏ. Since f(x)=1, we have ∫ f dμ=∫ f dν = ∫ 1 dμ = ∫ 1 dν.
3. Now, let's consider a continuous function f:Rᵏ→R with f(x)=0, where x is any point in Rᵏ. Since f(x)=0, we have ∫ f dμ=∫ f dν = ∫ 0 dμ = ∫ 0 dν.
4. From step 2 and step 3, we have ∫ 1 dμ = ∫ 1 dν and ∫ 0 dμ = ∫ 0 dν.
5. By subtracting the integrals in step 4, we have ∫ (1 - 0) dμ = ∫ (1 - 0) dν.
6. Simplifying the integrals, we get ∫ dμ = ∫ dν.
7. The integrals in step 6 represent the measures μ and ν.
8. Since ∫ dμ = ∫ dν for all continuous functions f:Rᵏ→R, we can conclude that μ=ν.