Integrals of Functions Vanishing or Agreeing off a Null Set on a Compact Interval

Let $a,b$ be real numbers with $a<b$ and adopt the notation $\mathcal{B}_{[a,b]}$ and $\lambda_{[a,b]}$ of the restricted Lebesgue measure space on a compact interval, so that $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ is a measure space by claim 1 there. Measurability of a real-valued function on $[a,b]$ means measurability with respect to $\mathcal{B}_{[a,b]}$ and the Borel $\sigma$-algebra of the real line, and integrability and the integral $\int_{[a,b]}f\,d\lambda_{[a,b]}$ are those of this measure space.

Call a set $N\in\mathcal{B}_{[a,b]}$ null if $\lambda_{[a,b]}(N)=0$.

Then the following hold.

1. (Vanishing off a null set.) Let $N$ be null and let $h:[a,b]\to\mathbb{R}$ be measurable with $h(t)=0$ for every $t\in[a,b]\setminus N$. Then $h$ is integrable and

$$\int_{[a,b]}h\,d\lambda_{[a,b]}=0 .$$

2. (Agreeing off a null set.) Let $N$ be null and let $f,g:[a,b]\to\mathbb{R}$ be integrable with $f(t)=g(t)$ for every $t\in[a,b]\setminus N$. Then

$$\int_{[a,b]}f\,d\lambda_{[a,b]}=\int_{[a,b]}g\,d\lambda_{[a,b]} .$$