Do I abuse notation ? $\mu(dx)=f(x)dx=f(g(t))g'(t)dt=\mu(dt)$?

Question

I have that $$\mu(A)=\int_A f(x)dx=\int_B f(g(t))g'(t)dt.$$

Do I abuse notation when I right $$\mu(dx)=f(x)dx=f(g(t))g'(t)dt=\mu(dt) \ \ ?$$

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In fact in an exercise of physic I have that $dF=F(x)dx$. Since $$F=\int F(x)dx=\int g(\theta )rd\theta , $$

we get $$dF=g(\theta )rd\theta.$$

I'm a bit confuse. Shouldn't it be $$dF(x)=F(x)dx=g(\theta )rd\theta =dF(t) \ \ ?$$

Could someone explain what we do ? (I can accept it in some way, but I would like to understand better).

Answer 1

While it is true that $\int f(x)d\mu(x) =\int f(t)d\mu(t)$ you cannot write $d\mu(x) =d\mu(t)$.